Integral Calculus IIT JEE

Amit M Agarwal

Chapter 3

Area of Bounded Regions - all with Video Answers

Educators

Chapter Questions

Problem 1

A Point $P(x, y)$ moves such that $[x+y+1]=[x]$. (where [.] denotes greatest integer function) and $x \in(0,2)$, then the area represented by all the possible positions of $P_{,}$ is
(a) $\sqrt{2}$
(b) $2 \sqrt{2}$
(c) $4 \sqrt{2}$
(d) 2

Vikash Ranjan

Vikash Ranjan

Numerade Educator

Problem 2

If $f:[-1,1] \rightarrow\left[-\frac{1}{2}, \frac{1}{2}\right], f(x)=\frac{x}{1+x^{2}}$, The area bounded
by $y=f^{-1}(x), X$ -axis, $x=\frac{1}{2}, x=-\frac{1}{2}$ is
(a) $\frac{1}{2} \log e$
(b) $\log \left(\frac{e}{2}\right)$
(c) $\frac{1}{2} \log \frac{e}{3}$
(d) $\frac{1}{2} \log \left(\frac{e}{2}\right)$

Vikash Ranjan

Numerade Educator

Problem 3

If the length of latusrectum of ellipse $E_{1}: 4(x+y-1)^{2}+2(x-y+3)^{2}=8$
and $E_{2}: \frac{x^{2}}{p}+\frac{y^{2}}{p^{2}}=1,(0<p<1)$ are equal, then area of ellipse $E_{2}$, is
(a) $\frac{\pi}{2}$
(b) $\frac{\pi}{\sqrt{2}}$
(c) $\frac{\pi}{2 \sqrt{2}}$
(d) $\frac{\pi}{4}$

Vikash Ranjan

Numerade Educator

Problem 4

The area of bounded by the curve $4\left|x-2017^{2017}\right|+5\left|y-2017^{2017}\right| \leq 20$, is
(a) 60
(b) 50
(c) 40
(d) 30

Vikash Ranjan

Numerade Educator

Problem 4

The area of bounded by the curve $4\left|x-2017^{2017}\right|+5\left|y-2017^{2017}\right| \leq 20$, is
(a) 60
(b) 50
(c) 40
(d) 30

Vikash Ranjan

Numerade Educator

Problem 5

If the area bounded by the corve $y=x^{2}+1, y=x$ and the pair of lines $x^{2}+y^{2}+2 x y-4 x-4 y+3=0$ is $K$ units, then the area of the region bounded by the curve $y=x^{2}+1, y=\sqrt{x-1}$ and the pair of lines $(x+y-1)(x+y-3)=0$, is
(a) $K$
(b) $2 \mathrm{~K}$
(c) $\frac{K}{2}$
(d) None of these

Vikash Ranjan

Numerade Educator

Problem 6

Suppose $y=f(x)$ and $y=g(x)$ are two functions whose graphs intersect at the three points $(0,4),(2,2)$ and $(4,0)$ with $f(x)>g(x)$ for $0<x<2$ and $f(x)<g(x)$ for $2<x<4 .$ If $\int_{0}^{4}[f(x)-g(x)] d x=10$
and $\int_{2}^{4}[g(x)-f(x)] d x=5$, then the area between two curves for $0<x<2$, is
(a) 5
(b) 10
(c) 15
(d) 20

Vikash Ranjan

Numerade Educator

Problem 7

Let ' $a$ ' be a positive constant number. Consider two curves $C_{1}: y=e^{x}, C_{2}: y=e^{a-x} .$ Let $S$ be the area of the part surrounding by $C_{1}, C_{2}$ and the $Y$ -axis, then $\lim _{a \rightarrow 0} \frac{S}{a^{2}}$
equals
(a) 4
(b) $1 / 2$
(c) 0
(d) $1 / 4$

Vikash Ranjan

Numerade Educator

Problem 8

3 points $O(0,0), P\left(a, a^{2}\right), Q\left(-b, b^{2}\right)(a>0, b>0)$ are on the parabola $y=x^{2} .$ Let $S_{1}$ be the area bounded by the line $P Q$ and the parabola and let $S_{2}$ be the area of the $\triangle O P Q$. the minimum value of $S_{1} / S_{2}$ is
(a) $4 / 3$
(b) $5 / 3$
(c) 2
(d) $7 / 3$

Vikash Ranjan

Numerade Educator

Problem 9

Area enclosed by the graph of the function $y=\ln ^{2} x-1$ lying in the 4 th quadrant is
(a) $\frac{2}{e}$
(b) $\frac{4}{e}$
(c) $2\left(e+\frac{1}{e}\right)$
(d) $4\left(e-\frac{1}{e}\right)$

Vikash Ranjan

Numerade Educator

Problem 10

The area bounded by $y=2-|2-x|$ and $y=\frac{3}{|x|}$ is
(a) $\frac{4+3 \ln 3}{2}$
(b) $\frac{19}{8}-3 \ln 2$
(c) $\frac{3}{2}+\ln 3$
(d) $\frac{1}{2}+\ln 3$

Vikash Ranjan

Numerade Educator

Problem 11

Suppose $g(x)=2 x+1$ and $h(x)=4 x^{2}+4 x+5$ and $h(x)=(f \circ g)(x)$. The area enclosed by the graph of the function $y=f(x)$ and the pair of tangents drawn to it from the origin, is
(a) $8 / 3$
(b) $16 / 3$
(c) $32 / 3$
(d) None of these

Vikash Ranjan

Numerade Educator

Problem 12

The area bounded by the curves $y=-\sqrt{-x}$ and $x=-\sqrt{-y}$ where $x, y \leq 0$
(a) cannot be determined
(b) is $1 / 3$
(c) is $2 / 3$
(d) is same as that of the figure bounded by the curves $y=\sqrt{-x} ; x \leq 0$ and $x=\sqrt{-y} ; y \leq 0$

Vikash Ranjan

Numerade Educator

Problem 13

$y=f(x)$ is a function which satisfies
(i) $f(0)=0$
(ii) $f^{*}(x)=f^{\prime}(x)$ and
(iii) $f^{\prime}(0)=1$
Then, the area bounded by the graph of $y=f(x)$, the lines $x=0, x-1=0$ and $y+1=0$ is
(a) $e$
(b) $e-2$
(c)e-1
(d) $e+1$

Vikash Ranjan

Numerade Educator

Problem 14

Area of the region enclosed between the curves $x=y^{2}-1$ and $x=|y| \sqrt{1-y^{2}}$ is
(a) 1
(b) $4 / 3$
(c) $2 / 3$
(d) 2

Vikash Ranjan

Numerade Educator

Problem 15

The area bounded by the curve $y=x e^{-x} ; x y=0$ and $x=c$ where $c$ is the $x$ -coordinate of the curve's inflection point, is
(a) $1-3 e^{-2}$
(b) $1-2 e^{-2}$
(c) $1-e^{-2}$
(d) 1

Vikash Ranjan

Numerade Educator

Problem 16

If $(a, 0) ; a>0$ is the point where the curve $y=\sin 2 x-\sqrt{3} \sin x$ cuts the $X$ -axis first, $A$ is the area bounded by this part of the curve, the origin and the positive $X$ -axis, then
(a) $4 A+8 \cos a=7$
(b) $4 A+8 \sin a=7$
(c) $4 A-8 \sin a=7$
(d) $4 A-8 \cos a=7$

Vikash Ranjan

Numerade Educator

Problem 17

The curve $y=a x^{2}+b x+c$ passes through the point $(1,2)$ and its tangent at origin is the line $y=x$. The area bounded by the curve, the ordinate of the curve at minima and the tangent line is
(a) $\frac{1}{24}$
(b) $\frac{1}{12}$
(c) $\frac{1}{8}$
(d) $\frac{1}{6}$

Vikash Ranjan

Numerade Educator

Problem 18

A function $y=f(x)$ satisfies the differential equation $\frac{d y}{d x}-y=\cos x-\sin x$, with initial condition that $y$ is bounded when $x \rightarrow \infty$, The area enclosed by $y=f(x), y=\cos x$ and the $Y$ -axis in the 1st quadrant
(a) $\sqrt{2}-1$
(b) $\sqrt{2}$
(c) 1
(d) $1 / \sqrt{2}$

Vikash Ranjan

Numerade Educator

Problem 19

If the area bounded between $X$ -axis and the graph of $y=6 x-3 x^{2}$ between the ordinates $x=1$ and $x=a$ is 19 sq units, then ' $a$ ' can take the value
(a) 4 or $-2$
(b) two values are in $(2,3)$ and one in $(-1,0)$
(c) two values one in $(3,4)$ and one in $(-2,-1)$
(d) None of the above

Vikash Ranjan

Numerade Educator

Problem 20

Area bounded by $y=f^{-1}(x)$ and tangent and normal drawn to it at the points with abscissae $\pi$ and $2 \pi$, where $f(x)=\sin x-x$ is(ii) $\frac{\pi^{2}}{2}-1$
(b) $\frac{\pi^{2}}{2}-2$
(c) $\frac{\pi^{2}}{2}-4$
(d) $\frac{\pi^{2}}{2}$

Vikash Ranjan

Numerade Educator

Problem 21

If $f(x)=x-1$ and $g(x)=\mid f(|x|)-2$, then the area bounded by $y=g(x)$ and the curve $x^{2}-4 y+8=0$ is equal to
(a) $\frac{4}{3}(4 \sqrt{2}-5)$
(b) $\frac{1}{3}(4 \sqrt{2}-3)$
(c) $\frac{8}{3}(4 \sqrt{2}-3)$
(d) $\frac{8}{3}(4 \sqrt{2}-5)$

Srilakshmi E K

Numerade Educator

Problem 22

Let $S=\left\{(x, y): \frac{y(3 x-1)}{x(3 x-2)}<0\right\}$
$S^{\prime}=\{(x, y) \in A \times B:-1 \leq A \leq 1,-1 \leq B \leq 1\}$
then the area of the region enclosed by all points in $S \cap S^{\prime}$ is
(a) 1
(b) 2
(c) 3
(d) 4

Vikash Ranjan

Numerade Educator

Problem 23

The area of the region bounded between the curves $y=e\|x|\ln | x\|, x^{2}+y^{2}-2(|x|+|y|)+1 \geq 0$ and $X$ -axis
where $|x| \leq 1$, if $\alpha$ is the $x$ -coordinate of the point of intersection of curves in 1st quadrant, is
(a) $4\left[\int_{0}^{\alpha} e x \ln x d x+\int_{a}^{1}\left(1-\sqrt{1-(x-1)^{2}}\right) d x\right]$
(b) $4\left[\int_{0}^{a} e x \ln x d x+\int_{1}^{a}\left(1-\sqrt{1-(x-1)^{2}}\right) d x\right]$
(c) $4\left[-\int_{0}^{a} e x \ln x d x+\int_{\alpha}^{1}\left(1-\sqrt{1-(x-1)^{2}}\right) d x\right]$
(d) $2\left[\int_{0}^{a} e x \ln x d x+\int_{1}^{a}\left(1-\sqrt{1-(x-1)^{2}}\right) d x\right]$

Vikash Ranjan

Numerade Educator

Problem 24

A point $P$ lying inside the curve $y=\sqrt{2 a x-x^{2}}$ is moving such that its shortest distance from the curve at any position is greater than its distance from $X$ -axis. The point $P$ enclose a region whose area is equal to
(a) $\frac{\pi a^{2}}{2}$
(b) $\frac{a^{2}}{3}$
(c) $\frac{2 a^{2}}{3}$
(d) $\left(\frac{3 \pi-4}{6}\right) a^{2}$

Vikash Ranjan

Numerade Educator

Problem 25

The triangle formed by the normal to the curve $f(x)=x^{2}-a x+2 a$ at the point $(2,4)$ and the coordinate axes lies in second quadrant, if its area is 2 sq units, then a can be
(a) 2
(b) $17 / 4$
(c) 5
(d) $19 / 4$

Vikash Ranjan

Numerade Educator

Problem 26

Let $f$ and $g$ be continuous function on $a \leq x \leq b$ and set $p(x)=\max \{f(x), g(x)\}$ and $q(x)=\min \{f(x), g(x)\}$,
then the area bounded by the curves $y=p(x), y=q(x)$
and the ordinates $x=a$ and $x=b$ is given by
(a) $\int_{4}^{b}|f(x)-g(x)| d x$
(b) $\int_{a}^{b}|p(x)-q(x)| d x$
(c) $\left.\int_{a}^{b} \mid f(x)-g(x)\right\} d x$
(d) $\int_{a}^{b}\{p(x)-q(x)\} d x$

Vikash Ranjan

Numerade Educator

Problem 27

The area bounded by the parabola $y=x^{2}-7 x+10$ and $X$ -axis equals
(a) area bounded by $y=-x^{2}+7 x-10$ and $X$ -axis
(b) $1 / 6$ sq units
(c) $5 / 6$ sq units
(d) $9 / 2 \mathrm{sq}$ units

Vikash Ranjan

Numerade Educator

Problem 28

Area bounded by the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ is equal to
(a) $6 \pi$ sq units
(b) $3 \pi$ sq units
(c) $12 \pi$ sq units
(d) area bounded by the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$

Vikash Ranjan

Numerade Educator

Problem 29

There is a curve in which the length of the perpendicular from the origin to tangent at any point is equal to abscissa of that point. Then,
(a) $x^{2}+y^{2}=2$ is one such curve
(b) $y^{2}=4 x$ is one such curve
(c) $x^{2}+y^{2}=2 c x$ (c parameters) are such curves
(d) there are no such curves

Kumareshwaran Rathinasabapathy

Kumareshwaran Rathinasabapathy

Numerade Educator

Problem 30

Statement I The area of the curve $y=\sin ^{2} x$ from 0 to $\pi$ will be more than that of the curve $y=\sin x$ from 0 to
$\pi$.
Statement II $x^{2}>x$, if $x>1$

Vikash Ranjan

Numerade Educator

Problem 31

Statement 1 The area bounded by the curves $y=x^{2}-3$ and $y=k x+2$ is least if $k=0$. Statement II The area bounded by the curves $y=x^{2}-3$ and $y=k x+2$ is $\sqrt{k^{2}+20}$

Vikash Ranjan

Numerade Educator

Problem 32

Statement I The area of region bounded parabola $y^{2}=4 x$ and $x^{2}=4 y$ is $\frac{32}{3}$ sq units. Statement II The area of region bounded by parabola $y^{2}=4 a x$ and $x^{2}=4 b y$ is $\frac{16}{3} a b$

Vikash Ranjan

Numerade Educator

Problem 33

Statement I The area by region $|x+y|+|x-y| \leq 2$ is 8 sq units. Statement II Area enclosed by region $|x+y|+|x-y| \leq 2$ is symmetric about $X$ -axis.

Vikash Ranjan

Numerade Educator

Problem 34

Statement I Area bounded by $y=x(x-1)$ and $y=x(1-x)$ is $\frac{1}{3}$
Statement II Area bounded by $y=f(x)$ and $y=g(x)$ is $\left|\int_{a}^{b}(f(x)-g(x)) d x\right|$ is true when $f(x)$ and $g(x)$ lies above $X$ -axis. (Where $a$ and $b$ are intersection of $y=f(x)$ and $y=g(x))$.

Vikash Ranjan

Numerade Educator

Problem 35

Then, the absolute area enclosed by $y=f(x)$ and $y=g(x)$ is given by
(a) $\sum_{r=0}^{n} \int_{x_{+}}^{x_{n}+1}(-1)^{\prime} \cdot h(x) d x$
(b) $\sum_{r=0}^{n} \int_{x,}^{x_{n}+1}(-1)^{r+1} \cdot h(x) d x$
(c) $2 \sum_{r=0}^{n} \int_{x_{4}}^{\pi_{4} x+1}(-1)^{\gamma} \cdot h(x) d x$
(d) $\frac{1}{2} \cdot \sum_{r=0}^{n} \int_{x_{4}}^{x_{r+1}}(-1)^{\gamma+1} h(x) d x$

Vikash Ranjan

Numerade Educator

Problem 36

In above inquestion the value of $n$, is
(a) $\underline{1}$
(b) $\underline{2}$
(c) 3
(d) 4

Vikash Ranjan

Numerade Educator

Problem 37

The whole area bounded by $y=f(x), y=g(x) x=0$ is
(a) $11 / 8$
(b) $8 / 3$
(c) $\underline{2}$
(d) $13 / 3$

Vikash Ranjan

Numerade Educator

Problem 38

Which of the following is true?
(a) $(2+a)^{2} f^{\prime \prime}(1)+(2-a)^{2} f^{\prime \prime}(-1)=0$
(b) $(2-a)^{2} f^{\prime \prime}(1)-(2+a)^{2} f^{\prime \prime}(-1)=0$
(c) $f^{\prime}(1) f^{\prime}(-1)=(2-a)^{2}$
(d) $f^{\prime}(1) f^{\prime}(-1)=-(2+a)^{2}$

Aayush Gupta

Numerade Educator

Problem 39

Which of the following is true?
(a) $f(x)$ is decreasing on $(-1,1)$ and has a local minimum at $x=1$
(b) $f(x)$ is increasing on $(-1,1)$ and has a local maximum at $x=1$

Hast Aggarwal

Numerade Educator

Problem 40

Let $g(x)=\int_{0}^{e^{x}} \frac{f^{\prime}(t)}{1+t^{2}} d t .$ Which of the following is true?
(a) $g^{\prime}(x)$ is positive on $(-\infty, 0)$ and negative on $(0, \infty)$
(b) $g^{\prime}(x)$ is negative on $(-\infty, 0)$ and positive on $(0, \infty)$
(c) $g^{\prime}(x)$ change sign on both $(-\infty, 0)$ and $(0, \infty)$
(d) $g^{\prime}(x)$ does not change sign on $(-\infty, \infty)$

Amy Jiang

Numerade Educator

Problem 41

The area enclosed by the asteroid $\left(\frac{x}{a}\right)^{2 / 3}+\left(\frac{y}{a}\right)^{2 / 3}=1$ is
(a) $\frac{3}{4} a^{2} \pi$
(b) $\frac{3}{18} \pi a^{2}$
(c) $\frac{3}{8} \pi a^{2}$
(d) $\frac{3}{4} a \pi$

Vikash Ranjan

Numerade Educator

Problem 42

The area of the region bounded by an arc of the cycloid $x=a(t-\sin t), y=a(1-\cos t)$ and the $X$ -axis is
(a) $6 \pi a^{2}$
(b) $3 \pi a^{2}$
(c) $4 \pi a^{2}$
(d) None of these

Vikash Ranjan

Numerade Educator

Problem 43

Area of the loop described as $x=\frac{t}{3}(6-t), y=\frac{t^{2}}{8}(6-t)$ is
(a) $\frac{27}{5}$
(b) $\frac{24}{5}$
(c) $\frac{27}{6}$
(d) $\frac{21}{5}$

Vikash Ranjan

Numerade Educator

Problem 44

Match the statements of Column I with values of Column II.

Hast Aggarwal

Numerade Educator

Problem 45

Match the following :
(A) Area enclosed by $y=|x|,|x|=1$ and
(p) $\quad 2$ $y=0$ is
(B) Area enclosed by the curve $y=\sin x$,
(q) 4 $x=0, x=\pi$ and $y=0$ is
(C) If the area of the region bounded by
(r) 27
$x^{2} \leq y$ and $y \leq x+2$ is $\frac{k}{4}$, then $k$ is equal to
(D) Area of the quadrilateral formed by
(s) 18
tangents at the ends of latusrectum of ellipse of ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$ is

James Kiss

Numerade Educator

Problem 46

Consider $f(x)=x^{2}-3 x+2$ The area bounded by $|y|=|f(|x|)|, x \geq 1$ is $A$, then find the value of $3 A+2$

Vikash Ranjan

Numerade Educator

Problem 47

The value of $c+2$ for which the area of the figure bounded by the curve $y=8 x^{2}-x^{5}$; the straight lines $x=1$ and $x=c$ and $X$ -axis is equal to $\frac{16}{3}$, is ..........

Aman Gupta

Numerade Educator

Problem 48

The area bounded by $y=2-|2-x| ; y=\frac{3}{|x|}$ is $\frac{k-3 \ln 3}{2}$, then $k$ is equal to .........

Vikash Ranjan

Numerade Educator

Problem 49

The area of the $\triangle A B C$, coordinates of whose vertices are $A(2,0), B(4,5)$ and $C(6,3)$ is .........

Vikash Ranjan

Numerade Educator

Problem 50

A point $P$ moves in $X Y$ -plane in such a way that $[|x|]+[|y|]=1$, where [ ] denotes the greatest integer function. Area of the region representing all possible of the point $P$ is equal to ............

Vikash Ranjan

Numerade Educator

Problem 51

Let $f:[0,1] \rightarrow\left[0, \frac{1}{2}\right]$ be a function such that $f(x)$ is a
polynomial of 2 nd degree, satisfy the following condition :
(a) $f(0)=0$
(b) has a maximum value of $\frac{1}{2}$ at $x=1$. If $A$ is the area bounded by $y=f(x) ; y=f^{-1}(x)$ and the line $2 x+2 y-3=0$ in 1st quadrant, then the value of $24 A$ is equal to ........ .

Anurag Kumar

Numerade Educator

Problem 52

Let $f(x)=\min \left\{\sin ^{-1} x, \cos ^{-1} x, \frac{\pi}{6}\right\}, x \in[0,1]$ If area
bounded by $y=f(x)$ and $X$ -axis, between the lines $x=0$ and $x=1$ is $\frac{a-X}{b(\sqrt{3}+1)}$. Then, $(a-b)$ is ............

Vikash Ranjan

Numerade Educator

Problem 53

Let $f$ be a real valued function satisfying $f\left(\frac{x}{y}\right)=f(x)-f(y)$ and $\lim _{x \rightarrow 0} \frac{f(1+x)}{x}=3$ Find the area
bounded by the curve $y=f(x)$, the $Y$ -axis and the line $y=3$ where $x, y \in R^{+}$.

Vikash Ranjan

Numerade Educator

Problem 54

Find a continuous function ' $f$ :
$\left(x^{4}-4 x^{2}\right) \leq f(x) \leq\left(2 x^{2}-x^{3}\right)$ such that the area
bounded by $y=f(x), y=x^{4}-4 x^{2}$, the $Y$ -axis and the line $x=t,(0 \leq t \leq 2)$ is $k$ times the area bounded by $y=f(x), y=2 x^{2}-x^{3}, Y$ -axis and line $x=t,(0 \leq t \leq 2)$

Vikash Ranjan

Numerade Educator

Problem 55

Let $f(t)=|t-1|-|t|+|t+1|, \forall t \in R$ and
$g(x)=\max \{f(t): x+1 \leq t \leq x+2\} ; \forall x \in R$. Find $g(x)$
and the area bounded by the curve $y=g(x)$, the $X$ -axis and the lines $x=-3 / 2$ and $x=5$.

Vikash Ranjan

Numerade Educator

Problem 56

Let $f(x)=$ minimum $\left\{e^{x}, 3 / 2,1+e^{-x}\right\}, 0 \leq x \leq 1$. Find the area bounded by $y=f(x), X$ -axis, $Y$ -axis and the line $x=1$

Vikash Ranjan

Numerade Educator

Problem 57

Find the area bounded by $y=f(x)$ and the curve $y=\frac{2}{1+x^{2}}$, where $f$ is a continuous function satisfying the conditions $f(x) \cdot f(y)=f(x y), \forall x, y \in R$
and $f^{\prime}(1)=2 f(1)=1$.

Vikash Ranjan

Numerade Educator

Problem 58

Find out the area bounded by the curve $y=\int_{1 / 8}^{\sin ^{2} x}\left(\sin ^{-1} \sqrt{t}\right) d t+\int_{1 / 8}^{\cos ^{2} x}\left(\cos ^{-1} \sqrt{t}\right) d t(0 \leq x \leq \pi / 2)$
and the curve satisfying the differential equation $y\left(x+y^{3}\right) d x=x\left(y^{3}-x\right) d y$ passing through $(4,-2)$

Gaurav Kalra

Numerade Educator

Problem 59

Let $T$ be an acute triangle. Inscribe a pair $R, S$ of rectangles in $T$ as shown :
$=$
Let $A(x)$ denote the area of polygon $X$ find the maximum value (or show that no maximum exists), of $\frac{A(R)+A(S)}{A(T)}$, where $T$ ranges over all triangles and $R, S$ over all rectangles as above.

Ankit Singh

Numerade Educator

Problem 60

Find the maximum area of the ellipse that can be inscribed in an isosceles triangles of area $A$ and having one axis lying along the perpendicular from the vertex of the triangles to its base.

Harshita Goel

Numerade Educator

Problem 61

In the adjacent figure the graphs of two function $y=f(x)$ and $y=\sin x$ are given. $y=\sin x$ intersects, $y=f(x)$ at $A(a, f(a)) ; B(\pi, 0)$ and $C(2 \pi, 0)$
$A_{i}(i=1,2,3)$ is the area bounded by the curves $y=f(x)$ and $y=\sin x$. between $x=0$ and $x=a ; i=1$
between $x=a$ and $x=\pi ; i=2$ between $x=\pi$ and $x=2 \pi ; i=3$. If $A_{1}=1-\sin a+(a-1) \cos a$, determine the function $f(x)$. Hence, determine $a$ and $A_{1}$. Also, calculate $A_{2}$ and $A_{3}$.

Vikash Ranjan

Numerade Educator

Problem 62

Find the area of the region bounded by curve $y=25^{x}+16$ and the curve $y=b .5^{x}+4$, whose tangent at the point $x=1$ make an angle $\tan ^{-1}(40 \ln 5)$ with the $X$ -axis.

Srilakshmi E K

Numerade Educator

Problem 63

If the circles of the maximum area inscribed in the region bounded by the curves $y=x^{2}-x-3$ and $y=3+2 x-x^{2}$, then the area of region $y-x^{2}+2 x+3 \leq 0, y+x^{2}-2 x-3 \leq 0$ and $s \leq 0$

Gaurav Kalra

Numerade Educator

Problem 64

Find limit of the ratio of the area of the triangle formed by the origin and intersection points of the parabola $y=4 x^{2}$ and the line $y=a^{2}$, to the area between the parabola and the line as $a$ approaches to zero.

Vikash Ranjan

Numerade Educator

Problem 65

Find the area of curve enclosed by :
$|x+y|+|x-y| \leq 4,|x| \leq 1, y \geq \sqrt{x^{2}-2 x+1}$

Vikash Ranjan

Numerade Educator

Problem 66

Calculate the area enclosed by the curve
$$
4 \leq x^{2}+y^{2} \leq 2(|x|+|y|)
$$

Vikash Ranjan

Numerade Educator

Problem 67

Find the area enclosed by the curve $[x]+[y]=4$ in 1 st quadrant (where [.] denotes greatest integer function).

Vikash Ranjan

Numerade Educator

Problem 68

Sketch the region and find the area bounded by the curves $|y+x| \leq 1,|y-x| \leq 1$ and $2 x^{2}+2 y^{2}=1$

Vikash Ranjan

Numerade Educator

Problem 69

Find the area of the region bounded by the curve, $2^{|x|}|y|+2^{|x|-1} \leq 1$, with in the square formed by the lines $|x| \leq 1 / 2|y| \leq 1 / 2$

Vikash Ranjan

Numerade Educator

Problem 70

Find all the values of the parameter $a(a \leq 1)$ for which the area of the figure bounded by the pair of straight lines $y^{2}-3 y+2=0$ and the curves $y=[a] x^{2}, y=\frac{1}{2}[a] x^{2}$ is
the greatest, where [.] denotes greatest integer function.

Angela Guo

Numerade Educator

Problem 71

If $f(x)$ is positive for all positive values of $X$ and $f^{\prime}(x)<0, f^{\prime \prime}(x)>0, \forall x \in R^{+}$, prove that
$\frac{1}{2} f(1)+\int_{1}^{n} f(x) d x<\sum_{r=1}^{n} f(r)<\int_{1}^{n} f(x) d x+f(1)$

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 72

Area of the region $\{(x, y)\} \in R^{2}: y \geq \sqrt{|x+3|}$,
$5 y \leq(x+9) \leq 15\}$ is equal to $\quad$ [Single Correct Option 2016]
(a) $\frac{1}{6}$
(b) $\frac{4}{3}$
(c) $\frac{3}{2}$
(d) $\frac{5}{3}$

Srilakshmi E K

Numerade Educator

Problem 73

Let $F(x)=\int_{x}^{x^{2}+\frac{\pi}{6}} 2 \cos ^{2} t d t$ for all $x \in R$ and
$f:\left[0, \frac{1}{2}\right] \rightarrow[0, \infty)$ be a continuous function. For $a \in\left[0, \frac{1}{2}\right]$, if $F^{\prime}(a)+2$ is the area of the region bounded by $x=0, y=0, y=f(x)$ and $x=a$, then $f(0)$ is

Vikash Ranjan

Numerade Educator

Problem 74

The common tangents to the circle $x^{2}+y^{2}=2$ and the parabola $y^{2}=8 x$ touch the circle at the points $P, Q$ and the parabola at the points $R, S$. Then, the area (in sq units) of the quadrilateral $P Q R S$ is

Aman Gupta

Numerade Educator

Problem 75

The area enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ over the interval $\left[0, \frac{\pi}{2}\right]$ is
[Single Correct Option 2014]
(a) $4(\sqrt{2}-1)$
(b) $2 \sqrt{2}(\sqrt{2}-1)$
(c) $2(\sqrt{2}+1)$
(d) $2 \sqrt{2}(\sqrt{2}+1)$

Gaurav Kalra

Numerade Educator

Problem 76

If $S$ be the area of the region enclosed by $y=e^{-x^{2}}, y=0, x=0$ and $x=1$. Then,
[More than One Option Correct 2012]
(a) $S \geq \frac{1}{e}$
(b) $S \geq 1-\frac{1}{e}$
(c) $S \leq \frac{1}{4}\left(1+\frac{1}{\sqrt{e}}\right)$
(d) $S \leq \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{e}}\left(1-\frac{1}{\sqrt{2}}\right)$

Srilakshmi E K

Numerade Educator

Problem 77

Let $f:[-1,2] \rightarrow[0, \infty)$ be a continuous function such that $f(x)=f(1-x), \forall x \in[-1,2]$ If $R_{1}=\int_{-1}^{2} x f(x) d x$ and $R_{2}$
are the area of the region bounded by $y=f(x)$ $x=-1, x=2$ and the $X$ -axis. Then, [Single Correct Option 2011]
(a) $R_{1}=2 R_{2}$
(b) $R_{1}=3 R_{2}$
(c) $2 R_{1}=R_{2}$
(d) $3 R_{1}=R_{2}$

Vikash Ranjan

Numerade Educator

Problem 78

If the straight line $x=b$ divide the area enclosed by $y=(1-x)^{2}, y=0$ and $x=0$ into two parts $R_{1}(0 \leq x \leq b)$ and $R_{2}(b \leq x \leq 1)$ such that $R_{1}-R_{2}=\frac{1}{4}$, Then, $b$ equals to $\quad$ [Single Correct Optlon 2011]
(a) $\frac{3}{4}$
(b) $\frac{1}{2}$
(c) $\frac{1}{3}$
(d) $\frac{1}{4}$

Vikash Ranjan

Numerade Educator

Problem 79

Area of the region bounded by the curve $y=e^{x}$ and lines $x=0$ and $y=e$ is [More than One Option Correct 2009]
(a) $e-1$
(b) $\int_{1}^{\theta} \ln (e+1-y) d y$
(c) $e-\int_{0}^{1} e^{x} d x$
(d) $\int_{1}^{x} \ln y d y$

Vikash Ranjan

Numerade Educator

Problem 80

The area of the region between the curves $y=\sqrt{\frac{1+\sin x}{\cos x}}$ and $y=\sqrt{\frac{1-\sin x}{\cos x}}$ and bounded by the
lines $x=0$ and $x=\frac{\pi}{4}$ is $\quad$ [Single Correct Option 2008]
(a) $\int_{0}^{\sqrt{2}-1} \frac{t}{\left(1+t^{2}\right) \sqrt{1-t^{2}}} d t$
(b) $\int_{0}^{\sqrt{2}-1} \frac{4 t}{\left(1+t^{2}\right) \sqrt{1-t^{2}}} d t$
(c) $\int_{0}^{\sqrt{2}+1} \frac{4 t}{\left(1+t^{2}\right) \sqrt{1-t^{2}}} d t$
(d) $\int_{0}^{\sqrt{2}+1} \frac{t}{\left(1+t^{2}\right) \sqrt{1-t^{2}}} d t$

Vikash Ranjan

Numerade Educator

Problem 81

If $f(-10 \sqrt{2})=2 \sqrt{2}$, then $f^{\prime \prime}(-10 \sqrt{2})$ is equal to
(a) $\frac{4 \sqrt{2}}{7^{3} 3^{2}}$
(b) $-\frac{4 \sqrt{2}}{7^{3} 3^{2}}$
(c) $\frac{4 \sqrt{2}}{7^{3} 3}$
(d) $-\frac{4 \sqrt{2}}{7^{\frac{1}{3}}}$

Vikash Ranjan

Numerade Educator

Problem 82

The area of the region bounded by the curve $y=f(x)$, the $X$ -axis and the lines $x=a$ and $x=b$, where $-\infty<a<b<-2$ is
(a) $\int_{a}^{b} \frac{x}{3\left[\{f(x)\}^{2}-1\right]} d x+b f(b)-a f(a)$
(b) $-\int_{a}^{b} \frac{x}{3\left[\{f(x)\}^{2}-1\right]} d x+b f(b)-a f(a)$
(c) $\int_{a}^{b} \frac{x}{3\left[\{f(x)\}^{2}-1\right]} d x-b f(b)+a f(a)$
(d) $-\int_{a}^{b} \frac{x}{3\left[\{f(x)\}^{2}-1\right]} d x-b f(b)+a f(a)$

Vikash Ranjan

Numerade Educator

Problem 83

$\int_{-1}^{1} g^{\prime}(x) d x$ is equal to
(a) $2 g(-1)$
(b) 0
(c) $-2 g(1)$
(d) $2 g(1)$

Srilakshmi E K

Numerade Educator

Problem 84

The area (in sq. units) of the region $\quad$ $\left\{(x, y): x \geq 0, x+y \leq 3, x^{2} \leq 4 y\right\}$ and $\left.y \leq 1+\sqrt{x}\right\}$ is
(a) $\frac{5}{2}$
(b) $\frac{59}{12}$
(c) $\frac{3}{2}$
(d) $\frac{7}{3}$

Srilakshmi E K

Numerade Educator

Problem 85

The area (in sq units) of the region $\left\{(x, y): y^{2} \geq 2 x\right.$ and $\left.x^{2}+y^{2} \leq 4 x, x \geq 0, y \geq 0\right\}$ is $\quad$
(a) $\pi-\frac{4}{3}$
(b) $\pi-\frac{8}{3}$
(c) $\pi-\frac{4 \sqrt{2}}{3}$
(d) $\frac{\pi}{2}-\frac{2 \sqrt{2}}{3}$

Vikash Ranjan

Numerade Educator

Problem 86

The area (in sq units) of the region described by $\left\{(x, y): y^{2} \leq 2 x\right.$ and $\left.y \geq 4 x-1\right\}$ is
(a) $\frac{7}{32}$
(b) $\frac{5}{64}$
(c) $\frac{15}{64}$
(d) $\frac{9}{32}$

Srilakshmi E K

Numerade Educator

Problem 87

The area (in sq units) of the quadrilateral formed by the tangents at the end points of the latusrectum to the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$ is $\quad$
(a) $\frac{27}{4}$
(b) 18
(c) $\frac{27}{2}$
(d) 27

Caleb Fink

Numerade Educator

Problem 88

The area of the region described by $A=\left\{(x, y): x^{2}+y^{2} \leq 1\right.$ and $\left.y^{2} \leq 1-x\right\}$ is
(a) $\frac{\pi}{2}+\frac{4}{3}$
(b) $\frac{\pi}{2}-\frac{4}{3}$
(c) $\frac{\pi}{2}-\frac{2}{3}$
(d) $\frac{\pi}{2}+\frac{2}{3}$

Vikash Ranjan

Numerade Educator

Problem 89

The area (in sq units) bounded by the curves $y=\sqrt{x}$, $2 y-x+3=0, X$ -axis and lying in the first quadrant is
(a) 9
(b) 36
(c) 18
(d) $\frac{27}{4}$

Vikash Ranjan

Numerade Educator

Problem 90

The area bounded between the parabolas $x^{2}=\frac{y}{4}$ and $x=e, y=\frac{1}{x}$ and the positive $X$ -axis is
(a) 1 sq unit
(b) $\frac{3}{2} \mathrm{sq}$ units
(c) $\frac{5}{2}$ sq units
(d) $\frac{1}{2}$ sq unit

Vikash Ranjan

Numerade Educator

Problem 91

The area of the region enclosed by the curves $y=x$, $x=e, y=\frac{1}{x}$ and the positive $X$ -axis is
(a) 1 sq unit
(b) $\frac{3}{2}$ sq units
(c) $\frac{5}{2}$ sq units
(d) $\frac{1}{2}$ sq unit

Vikash Ranjan

Numerade Educator

Problem 92

The area bounded by the curves $y=\cos x$ and $y=\sin x$
between the ordinates $x=0$ and $x=\frac{3 \pi}{2}$ is $\frac{21}{[2}$
(a) $(4 \sqrt{2}-2)$ sq units
(b) $(4 \sqrt{2}+2)$ sq units
(c) $(4 \sqrt{2}-1)$ sq units
(d) $(4 \sqrt{2}+1)$ sq units

Gaurav Kalra

Numerade Educator

Problem 93

The area of the region bounded by the parabola $(y-2)^{2}=x-1$, the tangent to the parabola at the point (2, 3) and the X-axis is
(a) 6 sq units
(b) 9 sq units
(c) 12 sq units
(d) 3 sq units

Vikash Ranjan

Numerade Educator

Problem 94

The area of the plane region bounded by the curves $x+2 y^{2}=0$ and $x+3 y^{2}=1$ is equal to
(a) $\frac{5}{3}$ sq units
(b) $\frac{1}{3}$ sq unit
(c) $\frac{2}{3}$ sq unit
(d) $\frac{4}{3} \mathrm{sq}$ units

Aman Gupta

Numerade Educator