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Calculus Early Transcendentals: Pearson New International Edition

Dale Varberg, Edwin J. Purcell, Steve E. Rigdon

Chapter 6

Applications of the Integral - all with Video Answers

Educators

Section 1

The Area of a Plane Region Volumes of Solids

02:27

Problem 1

Use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral for integrals) for the area of the indicated region.

Monica Miller
Monica Miller
Numerade Educator
02:27

Problem 2

Use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral for integrals) for the area of the indicated region.

Monica Miller
Monica Miller
Numerade Educator
02:27

Problem 3

Use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral for integrals) for the area of the indicated region.

Monica Miller
Monica Miller
Numerade Educator
02:27

Problem 4

Use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral for integrals) for the area of the indicated region.

Monica Miller
Monica Miller
Numerade Educator
02:27

Problem 5

Use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral for integrals) for the area of the indicated region.

Monica Miller
Monica Miller
Numerade Educator
02:27

Problem 6

Use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral for integrals) for the area of the indicated region.

Monica Miller
Monica Miller
Numerade Educator
02:27

Problem 7

Use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral for integrals) for the area of the indicated region.

Monica Miller
Monica Miller
Numerade Educator
02:27

Problem 8

Use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral for integrals) for the area of the indicated region.

Monica Miller
Monica Miller
Numerade Educator
02:27

Problem 9

Use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral for integrals) for the area of the indicated region.

Monica Miller
Monica Miller
Numerade Educator
02:27

Problem 10

Use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral for integrals) for the area of the indicated region.

Monica Miller
Monica Miller
Numerade Educator
03:11

Problem 11

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$y=3-\frac{1}{3} x^{2}, y=0$, between $x=0$ and $x=3$

Monica Miller
Monica Miller
Numerade Educator
04:41

Problem 12

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$y=5 x-x^{2}, y=0$, between $x=1$ and $x=3$

Monica Miller
Monica Miller
Numerade Educator
05:10

Problem 13

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$y=(x-4)(x+2), y=0$, between $x=0$ and $x=3$

Monica Miller
Monica Miller
Numerade Educator
05:35

Problem 14

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$y=x^{2}-4 x-5, y=0$, between $x=-1$ and $x=4$

Monica Miller
Monica Miller
Numerade Educator
04:12

Problem 15

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$y=\frac{1}{4}\left(x^{2}-7\right), y=0$, between $x=0$ and $x=2$

Monica Miller
Monica Miller
Numerade Educator
03:57

Problem 16

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$y=x^{3}, y=0$, between $x=-3$ and $x=3$

Monica Miller
Monica Miller
Numerade Educator
03:41

Problem 17

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$y=\sqrt[3]{x}, y=0$, between $x=-2$ and $x=2$

Monica Miller
Monica Miller
Numerade Educator
03:40

Problem 18

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$y=\sqrt{x}-10, y=0$, between $x=0$ and $x=9$

Monica Miller
Monica Miller
Numerade Educator
04:17

Problem 19

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$y=(x-3)(x-1), y=x$

Nicole Wood
Nicole Wood
Numerade Educator
03:40

Problem 20

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$y=\sqrt{x}, y=x-4, x=0$

Monica Miller
Monica Miller
Numerade Educator
03:47

Problem 21

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$y=x^{2}-2 x, y=-x^{2}$

Monica Miller
Monica Miller
Numerade Educator
04:24

Problem 22

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$y=x^{2}-9, y=(2 x-1)(x+3)$

Monica Miller
Monica Miller
Numerade Educator
03:49

Problem 23

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$x=8 y-y^{2}, x=0$

Monica Miller
Monica Miller
Numerade Educator
03:47

Problem 24

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$x=(3-y)(y+1), x=0$

Monica Miller
Monica Miller
Numerade Educator
06:13

Problem 25

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$x=-6 y^{2}+4 y, x+3 y-2=0$

Monica Miller
Monica Miller
Numerade Educator
05:48

Problem 26

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$x=y^{2}-2 y, x-y-4=0$

Monica Miller
Monica Miller
Numerade Educator
04:41

Problem 27

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$4 y^{2}-2 x=0,4 y^{2}+4 x-12=0$

Monica Miller
Monica Miller
Numerade Educator
05:02

Problem 28

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$x=4 y^{4}, x=8-4 y^{4}$

Monica Miller
Monica Miller
Numerade Educator
03:47

Problem 29

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$y=e^{2 x}, y=0$, between $x=0$ and $x=\ln 2$

Monica Miller
Monica Miller
Numerade Educator
04:41

Problem 30

Sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer.
$y=e^{x}, y=e^{-x}$, between $x=0$ and $x=1$

Monica Miller
Monica Miller
Numerade Educator
06:05

Problem 31

Sketch the region $R$ bounded by $y=x+6, y=x^{3}$, and $2 y+x=0 .$ Then find its area. Hint: Divide $R$ into two pieces.

Monica Miller
Monica Miller
Numerade Educator
08:17

Problem 32

Find the area of the triangle with vertices at $(-1,4)$, $(2,-2)$, and $(5,1)$ by integration.

Monica Miller
Monica Miller
Numerade Educator
04:02

Problem 33

An object moves along a line so that its velocity at time $t$ is $v(t)=3 t^{2}-24 t+36$ feet per second. Find the displacement and total distance traveled by the object for $-1 \leq t \leq 9 .$

Monica Miller
Monica Miller
Numerade Educator
04:49

Problem 34

Follow the directions of Problem 33 if $v(t)=\frac{1}{2}+\sin 2 t$ and the interval is $0 \leq t \leq 3 \pi / 2$

Monica Miller
Monica Miller
Numerade Educator
08:19

Problem 35

Starting at $s=0$ when $t=0$, an object moves along a line so that its velocity at time $t$ is $v(t)=2 t-4$ centimeters per second. How long will it take to get to $s=12 ?$ To travel a total distance of 12 centimeters?

Monica Miller
Monica Miller
Numerade Educator
05:04

Problem 36

Consider the curve $y=1 / x^{2}$ for $1 \leq x \leq 6$.
(a) Calculate the area under this curve.
(b) Determine $c$ so that the line $x=c$ bisects the area of part (a).
(c) Determine $d$ so that the line $y=d$ bisects the area of nart (a).

Nicole Wood
Nicole Wood
Numerade Educator
01:56

Problem 37

Find the area of the region in the first quadrant below $y=e^{-x}$ above $y=\frac{1}{2}$

Narayan Hari
Narayan Hari
Numerade Educator
04:46

Problem 38

Find the area of the region trapped between $y=x e^{-x^{2}}$ and $y=x / 4 .$ Hint: There are two separate regions.

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:06

Problem 39

Use the Parabolic Rule with $n=8$ to approximate the area of the region trapped between $y=1-e^{-x^{2}}$ and $y=e^{-x^{2}} .$

Grace Muhihu
Grace Muhihu
Numerade Educator
05:02

Problem 40

Use the Parabolic Rule with $n=8$ to approximate the area of the region trapped between $y=\ln (x+1)$ and $y=x / 4$. Hint: One point of intersection is obvious; the other you must approximate.

Monica Miller
Monica Miller
Numerade Educator
02:14

Problem 41

Calculate areas $A, B, C$, and $D$ in Figure 12. Check by calculating $A+B+C+D$ in one integration.

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
00:32

Problem 42

Prove Cavalieri's Principle. (Bonaventura Cavalieri (1598-1647) developed this principle in $1635 .$ ) If two regions have the same height at every $x$ in $[a, b]$, then they have the same area (see Figure 13).

Xiaomeng Zhang
Xiaomeng Zhang
Numerade Educator
04:24

Problem 43

Use Cavalieri's Principle (not integration; see Problem
42) to show that the shaded regions in Figure 14 have the same area.

Umar Sohail Qureshi
Umar Sohail Qureshi
Numerade Educator
07:00

Problem 44

Find the area of the region trapped between $y=\sin x$ and $y=\frac{1}{2}, 0 \leq x \leq 17 \pi / 6$.

Monica Miller
Monica Miller
Numerade Educator