Chapter 5, Integrals Video Solutions, Calculus: Early Transcendentals

Section 3

The Fundamental Theorem of Calculus

00:32

Problem 1

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

Amrita Bhasin

Numerade Educator

12:51

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 Trentman

Numerade Educator

08:18

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 Trentman

Numerade Educator

12:30

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 Trentman

Numerade Educator

08:25

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 Druch

Numerade Educator

05:30

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 Trentman

Numerade Educator

02:48

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 Hotta

Numerade Educator

01:31

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 Hotta

Numerade Educator

00:25

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 Bhasin

Numerade Educator

00:24

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 Bhasin

Numerade Educator

00:25

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 Bhasin

Numerade Educator

00:43

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 Bhasin

Numerade Educator

00:30

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 Bhasin

Numerade Educator

00:23

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 Bhasin

Numerade Educator

00:49

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 Bhasin

Numerade Educator

00:34

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 Bhasin

Numerade Educator

00:34

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 Bhasin

Numerade Educator

03:05

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 Trentman

Numerade Educator

04:53

Problem 19

Evaluate the integral.

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

Michelle Bartolo

Numerade Educator

00:31

Problem 20

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

00:26

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 Bhasin

Numerade Educator

02:28

Problem 22

Evaluate the integral.

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

Yuki Hotta

Numerade Educator

00:28

Problem 23

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

00:28

Problem 24

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

00:29

Problem 25

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

01:31

Problem 26

Evaluate the integral.

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

Yuki Hotta

Numerade Educator

00:40

Problem 27

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

01:53

Problem 28

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

Mary Wakumoto

Numerade Educator

00:39

Problem 29

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

00:32

Problem 30

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

00:35

Problem 31

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

00:28

Problem 32

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

00:26

Problem 33

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

00:53

Problem 34

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

00:29

Problem 35

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

00:50

Problem 36

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

00:47

Problem 37

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

00:38

Problem 38

Evaluate the integral.

$ \displaystyle \int^1_0 \cosh t \,dt $

Amrita Bhasin

Numerade Educator

02:04

Problem 39

Evaluate the integral.

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

Mary Wakumoto

Numerade Educator

00:45

Problem 40

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

00:25

Problem 41

Evaluate the integral.

$ \displaystyle \int^4_0 2^s \,ds $

Amrita Bhasin

Numerade Educator

00:48

Problem 42

Evaluate the integral.

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

Amrita Bhasin

Numerade Educator

01:03

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 Bhasin

Numerade Educator

00:45

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 Bhasin

Numerade Educator

00:46

Problem 45

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

$ y = \sqrt{x} $, $ y = 0 $, $ x = 4 $

Amrita Bhasin

Numerade Educator

00:36

Problem 46

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

$ y = x^3 $, $ y = 0 $, $ x = 1 $

Amrita Bhasin

Numerade Educator

04:06

Problem 47

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

$ y = 4 - x^2 $, $ y = 0 $

Leon Druch

Numerade Educator

00:40

Problem 48

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

$ y = 2x - x^2 $, $ y = 0 $

Amrita Bhasin

Numerade Educator

00:50

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[3]{x} $, $ 0 \le x \le 27 $

Amrita Bhasin

Numerade Educator

00:50

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 Bhasin

Numerade Educator

02:31

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 Trost

Numerade Educator

01:01

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 Bhasin

Numerade Educator

00:48

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 Bhasin

Numerade Educator

00:44

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 Bhasin

Numerade Educator

00:41

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 Bhasin

Numerade Educator

03:18

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 Hotta

Numerade Educator

02:10

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 Barnes

University of North Texas

00:31

Problem 58

What is wrong with the equation?

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

Amrita Bhasin

Numerade Educator

00:56

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 Bhasin

Numerade Educator

01:00

Problem 60

Find the derivative of the function.

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

Amrita Bhasin

Numerade Educator

00:28

Problem 61

Find the derivative of the function.

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

Amrita Bhasin

Numerade Educator

01:05

Problem 62

Find the derivative of the function.

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

Amrita Bhasin

Numerade Educator

View

Problem 63

Find the derivative of the function.

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

Ma. Theresa Alin

Numerade Educator

00:44

Problem 64

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

Amrita Bhasin

Numerade Educator

01:01

Problem 65

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

Amrita Bhasin

Numerade Educator

00:46

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 Bhasin

Numerade Educator

00:44

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 Bhasin

Numerade Educator

View

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 Maheshwari

Numerade Educator

00:34

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 Bhasin

Numerade Educator

06:04

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 Trentman

Numerade Educator

06:52

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 Lin

Numerade Educator

06:57

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 Lin

Numerade Educator

11:34

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 Trentman

Numerade Educator

09:37

Problem 74

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

Chris Trentman

Numerade Educator

00:43

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 Bhasin

Numerade Educator

02:51

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 Barnes

University of North Texas