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### Problem 1

Explain exactly what is meant by the statement that "differentiation and integration are inverse processes." Amrita B.

### Problem 2

Let $\displaystyle g(x) = \int^x_0 f(t) \,dt$, where $f$ is the function whose graph is shown.

(a) Evaluate $g(x)$ for $x$ = 0, 1, 2, 3, 4, 5, and 6.
(b) Estimate $g(7)$.
(c) Where does $g$ have a maximum value? Where does it have a minimum value?
(d) Sketch a rough graph of $g$. Chris T.

### Problem 3

Let $g(x)=\int_{0}^{x} f(t) d t,$ where $f$ is the function whose graph is shown.
(a) Evaluate $g(x)$ for $x=0,1,2,3,4,5,$ and 6 .
(b) Estimate $g(7)$
(c) Where does $g$ have a maximum value? Where does it have a minimum value?
(d) Sketch a rough graph of $g$. Chris T.

### Problem 4

Let $\displaystyle g(x) = \int^x_0 f(t) \,dt$, where $f$ is the function whose graph is shown.

(a) Evaluate $g(0)$ and $g(6)$
(b) Estimate $g(x)$ for $x$ = 1, 2, 3, 4, and 5.
(c) On what interval is $g$ increasing?
(d) Where does $g$ have a maximum value?
(e) Sketch a rough graph of $g$.
(f) Use the graph in part (e) to sketch the graph of $g'(x)$. Compare with the graph of $f$. Chris T.

### Problem 5

Sketch the area represented by $g(x)$. Then find $g'(x)$ in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating.

$\displaystyle g(x) = \int^x_1 t^2\,dt$ Leon D.

### Problem 6

Sketch the area represented by $g(x)$. Then find $g'(x)$ in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating.

$\displaystyle g(x) = \int^x_0 (2 + \sin t)\,dt$ Chris T.

### Problem 7

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$\displaystyle g(x) = \int^x_0 \sqrt{t + t^3} \,dt$ Yuki H.

### Problem 8

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$\displaystyle g(x) = \int^x_1 \ln (1 + t^2) \,dt$ Yuki H.

### Problem 9

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$\displaystyle g(s) = \int^s_5 (t - t^2)^8 \,dt$ Amrita B.

### Problem 10

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$\displaystyle h(u) = \int^u_0 \frac{\sqrt{t}}{t + 1} \,dt$ Amrita B.

### Problem 11

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$\displaystyle F(x) = \int^0_x \sqrt{1 + \sec t} \,dt$

$$\biggl[ \textit{Hint:} \int^0_x \sqrt{1 + \sec t} \,dt = - \int^x_0 \sqrt{1 + \sec t} \,dt \biggr]$$ Amrita B.

### Problem 12

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$\displaystyle R(y) = \int^2_y t^3 \sin t \,dt$ Amrita B.

### Problem 13

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$\displaystyle h(x) = \int^{e^x}_1 \ln t \,dt$ Amrita B.

### Problem 14

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$\displaystyle h(x) = \int^{\sqrt{x}}_1 \frac{z^2}{z^4 + 1} \,dz$ Amrita B.

### Problem 15

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$\displaystyle y = \int^{3x + 2}_1 \frac{t}{1 + t^3} \,dt$ Amrita B.

### Problem 16

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$\displaystyle y = \int^{x^4}_0 \cos^2 \theta \,d\theta$ Amrita B.

### Problem 17

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$\displaystyle y = \int^{\pi/4}_{\sqrt{x}} \theta \tan \theta \,d\theta$ Amrita B.

### Problem 18

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$\displaystyle y = \int^1_{\sin x} \sqrt{1 + t^2} \,dt$ Chris T.

### Problem 19

Evaluate the integral.

$\displaystyle \int^3_1 (x^2 + 2x - 4) \,dx$

MB
Michelle B.

### Problem 20

Evaluate the integral.

$\displaystyle \int^1_{-1} x^{100} \,dx$ Amrita B.

### Problem 21

Evaluate the integral.

$\displaystyle \int^2_0 \biggl(\frac{4}{5}t^3 - \frac{3}{4}t^2 + \frac{2}{5}t \biggr) \,dt$ Amrita B.

### Problem 22

Evaluate the integral.

$\displaystyle \int^1_0 (1 - 8v^3 + 16v^7) \,dv$ Yuki H.

### Problem 23

Evaluate the integral.

$\displaystyle \int^9_1 \sqrt{x} \,dx$ Amrita B.

### Problem 24

Evaluate the integral.

$\displaystyle \int^8_1 x^{-2/3} \,dx$ Amrita B.

### Problem 25

Evaluate the integral.

$\displaystyle \int^{\pi}_{\pi/6} \sin \theta \,d\theta$ Amrita B.

### Problem 26

Evaluate the integral.

$\displaystyle \int^5_{-5} e \,dx$ Yuki H.

### Problem 27

Evaluate the integral.

$\displaystyle \int^1_0 (u + 2) (u - 3) \,du$ Amrita B.

### Problem 28

Evaluate the integral.
$\int_{0}^{4}(4-t) \sqrt{t} d t$ Amrita B.

### Problem 29

Evaluate the integral.

$\displaystyle \int^4_1 \frac{2 + x^2}{\sqrt{x}} \,dx$ Amrita B.

### Problem 30

Evaluate the integral.

$\displaystyle \int^2_{-1} (3u - 2)(u + 1) \,du$ Amrita B.

### Problem 31

Evaluate the integral.

$\displaystyle \int^{\pi/2}_{\pi/6} \csc t \cot t \,dt$ Amrita B.

### Problem 32

Evaluate the integral.

$\displaystyle \int^{\pi/3}_{\pi/4} \csc^2 \theta \,d\theta$ Amrita B.

### Problem 33

Evaluate the integral.

$\displaystyle \int^1_0 (1 + r)^3 \,dr$ Amrita B.

### Problem 34

Evaluate the integral.

$\displaystyle \int^3_0 (2\sin x - e^x) \,dx$ Amrita B.

### Problem 35

Evaluate the integral.

$\displaystyle \int^2_1 \frac{v^3 + 3v^6}{v^4} \,dv$ Amrita B.

### Problem 36

Evaluate the integral.

$\displaystyle \int^{18}_1 \sqrt{\frac{3}{z}} \,dz$ Amrita B.

### Problem 37

Evaluate the integral.

$\displaystyle \int^1_0 (x^e + e^x) \,dx$ Amrita B.

### Problem 38

Evaluate the integral.

$\displaystyle \int^1_0 \cosh t \,dt$ Amrita B.

### Problem 39

Evaluate the integral.

$\displaystyle \int^{\sqrt{3}}_{1/\sqrt{3}} \frac{8}{1 + x^2} \,dx$ Amrita B.

### Problem 40

Evaluate the integral.

$\displaystyle \int^3_1 \frac{y^3 - 2y^2 - y}{y^2} \,dy$ Amrita B.

### Problem 41

Evaluate the integral.

$\displaystyle \int^4_0 2^s \,ds$ Amrita B.

### Problem 42

Evaluate the integral.

$\displaystyle \int^{1/\sqrt{2}}_{1/2} \frac{4}{\sqrt{1 - x^2}} \,dx$ Amrita B.

### Problem 43

Evaluate the integral.

$\displaystyle \int^{\pi}_0 f(x) \,dx$ where $f(x) = \left\{ \begin{array}{ll} \sin x & \mbox{if$ 0 \le x < \pi/2 $}\\ \cos x & \mbox{if$ \pi/2 \le x \le \pi $} \end{array} \right.$ Amrita B.

### Problem 44

Evaluate the integral.

$\displaystyle \int^2_{-2} f(x) \,dx$ where $f(x) = \left\{ \begin{array}{ll} 2 & \mbox{if$ -2 \le x \le 0 $}\\ 4 - x^2 & \mbox{if$ 0 < x \le 2 $} \end{array} \right.$ Amrita B.

### Problem 45

Sketch the region enclosed by the given curves and calculate its area.

$y = \sqrt{x}$, $y = 0$, $x = 4$ Amrita B.

### Problem 46

Sketch the region enclosed by the given curves and calculate its area.

$y = x^3$, $y = 0$, $x = 1$ Amrita B.

### Problem 47

Sketch the region enclosed by the given curves and calculate its area.

$y = 4 - x^2$, $y = 0$ Leon D.

### Problem 48

Sketch the region enclosed by the given curves and calculate its area.

$y = 2x - x^2$, $y = 0$ Amrita B.

### Problem 49

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

$y = \sqrt{x}$, $0 \le x \le 27$ Amrita B.

### Problem 50

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

$y = x^{-4}$, $1 \le x \le 6$ Amrita B.

### Problem 51

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

$y = \sin x$, $0 \le x \le \pi$ Jacquelyn T.

### Problem 52

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

$y = \sec^2 x$, $0 \le x \le \pi/3$ Amrita B.

### Problem 53

Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch.

$\displaystyle \int^2_{-1} x^3 \,dx$ Amrita B.

### Problem 54

Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch.

$\displaystyle \int^{2\pi}_{\pi/6} \cos x \,dx$ Amrita B.

### Problem 55

What is wrong with the equation?

$\displaystyle \int^1_{-2} x^{-4} \, dx = \frac{x^{-3}}{-3} \Bigg]^1_{-2} = -\frac{3}{8}$ Amrita B.

### Problem 56

What is wrong with the equation?

$\displaystyle \int^2_{-1} \frac{4}{x^3} \, dx = -\frac{2}{x^2} \Bigg]^2_{-1} = -\frac{3}{2}$ Yuki H.

### Problem 57

What is wrong with the equation?

$\displaystyle \int^{\pi}_{\pi/3} \sec \theta \tan \theta \, d\theta = \sec \theta \Bigg]^{\pi}_{\pi/3} = -3$ Bobby B.
University of North Texas

### Problem 58

What is wrong with the equation?

$\displaystyle \int^{\pi}_{0} \sec^2 x \, dx = \tan x \Bigg]^{\pi}_{0} = 0$ Amrita B.

### Problem 59

Find the derivative of the function.

$g(x) = \displaystyle \int^{3x}_{2x} \frac{u^2 - 1}{u^2 + 1} \, du$

$\displaystyle \Biggl[ Hint: \int^{3x}_{2x} f(u) \, du = \int^0_{2x} f(u) \, du + \int^{3x}_0 f(u) \, du \Biggr]$ Amrita B.

### Problem 60

Find the derivative of the function.

$g(x) = \displaystyle \int^{1 + 2x}_{1 - 2x} t \sin t \, dt$ Amrita B.

### Problem 61

Find the derivative of the function.

$F(x) = \displaystyle \int^{x^2}_{x} e^{t^2} \, dt$ Amrita B.

### Problem 62

Find the derivative of the function.

$F(x) = \displaystyle \int^{2x}_{\sqrt{x}} \arctan t \, dt$ Amrita B.

### Problem 63

Find the derivative of the function.

$y = \displaystyle \int^{\sin x}_{\cos x} \ln (1 + 2v) \, dv$ Ma. Theresa A.

### Problem 64

If $\displaystyle f(x) = \int^x_0 (1 - t^2) e^{t^2} \,dt$, on what interval is $f$ increasing? Amrita B.

### Problem 65

On what interval is the curve $$y = \displaystyle \int^x_0 \frac{t^2}{t^2 + t + 2} \, dt$$ concave downward? Amrita B.

### Problem 66

Let $\displaystyle F(x) = \int^x_1 f(t) \, dt$, where $f$ is the function whose graph is shown. Where is $F$ concave downward? Amrita B.

### Problem 67

Let $\displaystyle F(x) = \int^x_2 e^{t^2} \, dt$. Find an equation of the tangent line to the curve $y = F(x)$ at the point with $x$-coordinate 2. Amrita B.

### Problem 68

If $\displaystyle f(x) = \int^{\sin x}_0 \sqrt{1 + t^2} \, dt$ and $\displaystyle g(y) = \int^y_3 f(x) \, dx$, find $g''(\pi/6)$. Darshan M.

### Problem 69

If $f(1) = 12$, $f'$ is continuous, and $\displaystyle \int^4_1 f'(x) \, dx = 17$, what is the value of $f(4)$? Amrita B.

### Problem 70

The error function $$\displaystyle \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int^x_0 e^{-t^2} \, dt$$ is used in probability, statistics, and engineering.
(a) Show that $\displaystyle \int^b_a e^{-t^2} \, dt = \frac{1}{2} \sqrt{\pi} [ \text{erf}(b) - \text{erf}(a) ]$.
(b) Show that the function $y = e^{x^2} \text{erf}(x)$ satisfies the differential equation $y' = 2xy + 2/\sqrt{\pi}$. Chris T.

### Problem 71

The Fresnel function $S$ was defined in Example 3 and graphed in Figures 7 and 8.

(a) At what values of $x$ does this function have local maximum values?
(b) On what intervals is the function concave upward?
(c) Use a graph to solve the following equation correct to two decimal places:
$$\displaystyle \int^x_0 \sin (\pi t^2/2) \, dt = 0.2$$ Frank L.

### Problem 72

The sine integral function $$\displaystyle \text{Si}(x) = \int^x_0 \frac{\sin t}{t} \, dt$$ is important in electrical engineering. [The integrand $f(t) = (\sin t)/t$ is not defined when $t = 0$, but we know that its limit is 1 when $t \to 0$. So we defined $f(0) = 1$ and this makes $f$ a continuous function everywhere.]

(a) Draw the graph of $\text{Si}$.
(b) At what values of $x$ does this function have local maximum values?
(c) Find the coordinates of the first inflection point to the right of the origin.
(d) Does this function have horizontal asymptotes?
(e) Solve the following equation correct to one decimal place:
$$\displaystyle \int^x_0 \frac{\sin t}{t} \, dt = 1$$ Frank L.

### Problem 73

Let $\displaystyle g(x) = \int^x_0 f(t) \, dt$, where $f$ is the function whose graph is shown.

(a) At what values of $x$ do the local maximum and minimum values of $g$ occur?
(b) Where does $g$ attain its absolute maximum value?
(c) On what intervals is $g$ concave downward?
(d) Sketch the graph of $g$. Chris T.

### Problem 74

Let $\displaystyle g(x) = \int^x_0 f(t) \, dt$, where $f$ is the function whose graph is shown.

(a) At what values of $x$ do the local maximum and minimum values of $g$ occur?
(b) Where does $g$ attain its absolute maximum value?
(c) On what intervals is $g$ concave downward?
(d) Sketch the graph of $g$. Chris T.

### Problem 75

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on $[0, 1]$.

$\displaystyle \lim_{n \to \infty} \sum_{i = 1}^n \biggl( \frac{i^4}{n^5} + \frac{i}{n^2} \biggr)$ Amrita B.

### Problem 76

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on $[0, 1]$.

$\displaystyle \lim_{n \to \infty} \frac{1}{n} \biggl( \sqrt{\frac{1}{n}} + \sqrt{\frac{2}{n}} + \sqrt{\frac{3}{n}} + \cdots + \sqrt{\frac{n}{n}} \biggr)$ Bobby B.
University of North Texas

### Problem 77

Justify (3) for the case $h < 0$. Amrita B.

### Problem 78

If $f$ is continuous and $g$ and $h$ are differentiable functions, find a formula for
$$\displaystyle \frac{d}{dx} \int^{h(x)}_{g(x)} f(t) \, dt$$ Amrita B.

### Problem 79

(a) Show that $1 \le \sqrt{1 + x^3} \le 1 + x^3$ for $x \ge 0$.
(b) Show that $\displaystyle 1 \le \int^1_0 \sqrt{1 + x^3} \, dx \le 1.25$ Chris T.

### Problem 80

(a) Show that $\cos (x^2) \ge \cos x$ for $0 \le x \le 1$.
(b) Deduce that $\displaystyle \int^{\pi/6}_0 \cos (x^2) \, dx \ge \frac{1}{2}$. Chris T.

### Problem 81

Show that $$0 \le \int^{10}_5 \frac{x^2}{x^4 + x^2 + 1} \, dx \le 0.1$$ by comparing the integrand to a simpler function. Amrita B.

### Problem 82

Let $f(x) = \left\{ \begin{array}{ll} 0 & \mbox{if$ x < 0 $}\\ x & \mbox{if$ 0 \le x \le 1 $}\\ 2 - x & \mbox{if$ 1 < x \le 2 $}\\ 0 & \mbox{if$ x > 2 $} \end{array} \right.$
and $$g(x) = \int^x_0 f(t) \, dt$$

(a) Find an expression for $g(x)$ similar to the one for $f(x)$.
(b) Sketch the graphs of $f$ and $g$.
(c) Where is $f$ differentiable? Where is $g$ differentiable? Chris T.

### Problem 83

Find a function $f$ and a number $a$ such that
$\displaystyle 6 + \int^x_a \frac{f(t)}{t^2} \, dt = 2 \sqrt{x}$ for all $x > 0$ Amrita B.

### Problem 84

The area labeled $B$ is three times the area labeled $A$. Express $b$ in terms of $a$. Aparna S.

### Problem 85

A manufacturing company owns a major piece of equipment that depreciates at the (continuous) rate $f = f(t)$, where $t$ is the time measured in months since its last overhaul. Because a fixed cost $A$ is incurred each time the machine is overhauled, the company wants to determine the optimal time $T$ (in months) between overhauls.

(a) Explain why $\displaystyle \int^t_0 f(s) ds$ represents the loss in value of the machine over the period of time $t$ since the last overhaul.
(b) Let $C = C(t)$ be given by $$C(t) = \frac{1}{t} \Biggl[ A + \int^t_0 f(s) ds \Biggr]$$
What does $C$ represent and why would the company want to minimize $C$?
(c) Show that $C$ has a minimum value at the numbers $t = T$ where $C(T) = f(T)$. Chris T.

### Problem 86

A high-tech company purchases a new computing system whose initial value is $V$. The system will depreciate at the rate $f = f(t)$ and will accumulate maintenance costs at the rate $g = g(t)$, where $t$ is the time measured in months. The company wants to determine the optimal time to replace the system.
(a) Let $$C(t) = \frac{1}{t} \int^t_0 [f(s) + g(s)] \, ds$$
Show that the critical numbers of $C$ occur at the numbers $t$ where $C(t) = f(t) + g(t)$.
(b) Suppose that
$f(t) = \left\{ \begin{array}{ll} \frac{V}{15} - \frac{V}{450}t & \mbox{if$ 0 < t \le 30 $}\\ 0 & \mbox{if$ t > 30 $} \end{array} \right.$
and $g(t) = \frac{Vt^2}{12,900}$ $t > 0$
Determine the length of time $T$ for the total depreciation $\displaystyle D(t) = \int^t_0 f(s) \, ds$ to equal the initial value $V$.
(c) Determine the absolute minimum of $C$ on $(0, T]$.
(d) Sketch the graphs of $C$ and $f + g$ in the same coordinate system, and verify the result in part (a) in this case. Chris T.