Chapter Questions
If $f(x)=f(2 x-x)$, then $\int_{0}^{2=1} f(x) d x$ is equal to(a) $\int_{a}^{0} f(2 a-x) d x \quad$ (b) $2 \int_{0}^{0} f(x) d x \quad$ (c) $-2 \int_{0}^{a} f(x) d x \quad(d)$
$\int_{0}^{n / 2} \frac{\sqrt{\operatorname{ain} x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$ in equal to(a) 0(b) $\underline{1}$(c) $\frac{\pi}{4}$$($ d $) \frac{5}{2}$.
The value of definite integral $\int_{-2}^{a}|x| d x$ is equal to(c) $\boldsymbol{a}$(b) $a^{2}$(c) $\underline{0}$(d) $2 \mathrm{a}$.
$\lim _{n \rightarrow-}\left[\frac{n}{n^{2}}+\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\ldots+\frac{n}{n^{2}+(n-1)^{2}}\right]$ is equal to$(a)-\frac{\pi}{4}$(b) 0(c) $\frac{\pi}{4}$(d) $\frac{\pi}{3}$
\begin{aligned}&\int_{0}^{\pi / 2} \frac{\cos 2 x}{\cos x+\sin x} d x \text { equals }\\&\begin{array}{llll}(\text { a })-1 & \text { (b) } 0 & \text { (c) } 1 & \text { (d) } 2\end{array}\end{aligned}
$\lim _{n \rightarrow-}\left(\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{3 n}\right)$ equals(a) log, 2(b) $2 \log _{e} 2$(c) log, 3(d) $2 \log _{y} 3$.
$\int_{0}^{\pi} \sin ^{5}\left(\frac{x}{2}\right)$ is equal to(a) $\frac{16}{15}$(b) $\frac{15}{16} \pi$(c) $\frac{16}{15} \pi^{2}$(d) $\frac{15}{16}$.
$\int_{0}^{\pi / 2} \sin ^{99} x \cos x d x$ is equal to(a) $\frac{1}{99}$(b) $\frac{\pi}{100}$(c) $\frac{99}{100}$(d) None of these.
The value of $\int_{-x / 2}^{\pi / 2} \cos ^{7} x d x$ is(a) $\frac{32 \pi}{35}$(b) $\frac{32}{35}$(c) zero.
The length of the arc of the equiangular spiral $r=a e^{\text {end e }}$ between the points for which the radii vectors are $r_{1}$ and $r_{2}$ is(a) $\left(r_{3}-r_{1}\right) \operatorname{cosec} \alpha$(b) $\left(r_{3}-r_{1}\right) \cos \alpha$(c) $\left(r_{2}-r_{1}\right) \sin \alpha$$\left(\right.$ d) $\left(r_{3}-r_{1}\right)$ nec $\alpha$.
The area of the region in the firet quadrant bounded by the $y$-axis and the curves $y=\sin x$ and $y=\cos x$ is(a) $\sqrt{2}$(b) $\sqrt{2}+1$(c) $\sqrt{2}-1$(d) $2 \sqrt{2}-1$
The value of $\int_{0}^{1} x^{3 / 2}(1-x)^{3 / 2} d x$ is(a) $\pi=32$(b) $-\pi / 32$$\begin{array}{ll}\text { (c) } 3 \pi / 128 & (d)-3 \pi / 128\end{array}$
The entire length of the cardioid $r=5(1+\cos \theta)$ is(a) 40(b) 30(c) 20(d) $\underline{5}$.
\text { The aren of the cardioid } r=a(1-\cos \theta) \text { is wmw }
If $S_{1}$ and $S_{2}$ are surface areas of the solids generated by revolving the ellipees $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ and $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1$ about the $y$-axis, then(a) $S_{1}>S_{j}$(b) $S_{1}<\mathcal{S}_{2}$(c) $S_{1}=S_{2}$(d) can't predict
\text { The area of the loop of the curve } r=a \sin 3 \theta \text { is }
\text { If } I_{n}=\int_{0}^{\pi / 4} \tan ^{n} \theta d \theta, \text { then } n\left(l_{n-1}+i_{m+1}\right)=\ldots
$\int_{0}^{2} x^{3} \sqrt{\left(2 x-x^{2}\right)} d x=$
$\int_{0}^{x / 2} \sin 2 \theta \log \tan \theta d \theta$ is equal te(a) 1(b) $-1$(c) 0(d) $\pi / 2$,
The area of the loop of the folium of Dencartes $x^{3}+y^{3}-3 x y=0$ is(a) $\pi$(b) $\pi / 2$(c) $1.5$$(d) \leq$
'The volume of the frustrum of a right cireular cene whose lower base has radius $r_{1}$ and upper base has radius $r_{2}$ and altitude is $h=\ldots \ldots$
The length of the are of the curve $y=$ log sec $x$ from $x=0$ te $x=\pi / 4$ is(a) $\log , 2$(b) $\log _{y} 3$(c) $\log _{4}(1+\sqrt{2})$(d) $\log _{4}(1+\sqrt{3})$.
If $v_{1}=$ volume of the solid generated by revolving the area included between $x$-axis and $x^{2}+y^{2}=a^{2}$ about $x$-axis:$v_{2}$ = velume of the solid generated by revolving the entire area of the circle $x^{2}+y^{2}=c^{2}$ about $x$-axis, then(a) $v_{1}=v_{2}$(b) $v_{2}=2 v_{1}$(c) $v_{4}=4 v_{1}$(d) $v_{2}=16 v_{r}$
If $f(r, \theta)=f(-r, \theta)$, then the curve is symmetrical about the... $\begin{array}{llll}\text { (a) initial line } & \text { (b) pole } & \text { (c) origin } & \text { (d) tangential line }\end{array}$
The volume generated by the revolution of the eurve $y=a^{2}\left(a^{2}+x^{2}\right)^{-1}$ about its asymptote is(a) $\pi^{2} a^{3} / 2$(b) $\operatorname{ra}^{3} \cdot \hat{2}$(c) $\pi a^{2} / 2$(d) $\operatorname{maf} 2$