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Student Solutions Manual for Stewart's / Single Variable Calculus: Early Transcendentals

James Stewart

Chapter 5

Integrals - all with Video Answers

Educators

Section 1

Areas and Distances

09:57

Problem 1

(a) By reading values from the given graph of $f$, use five rectangles to find a lower estimate and an upper estimate for the area under the given graph of $f$ from $x=0$ to $x=10$. In each case sketch the rectangles that you use.
(b) Find new estimates using 10 rectangles in each case.
GRAPH CANT COPY

Payton Sawyer
Payton Sawyer
Numerade Educator
17:14

Problem 2

(a) Use six rectangles to find estimates of each type for the area under the given graph of $f$ from $x=0$ to $x=12$.
(i) $L_6 \quad$ (sample points are left endpoints)
(ii) $R_6$ (sample points are right endpoints)
(iii) $M_6$ (sample points are midpoints)
(b) Is $L_6$ an underestimate or overestimate of the true area?
(c) Is $R_6$ an underestimate or overestimate of the true area?
(d) Which of the numbers $L_6, R_6$, or $M_6$ gives the best estimate? Explain.
GRAPH CANT COPY

EW
Eleanor Wang
Numerade Educator
04:10

Problem 3

(a) Estimate the area under the graph of $f(x)=1 / x$ from $x=1$ to $x=5$ using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?
(b) Repeat part (a) using left endpoints.

Gregory Higby
Gregory Higby
Numerade Educator
02:42

Problem 4

(a) Estimate the area under the graph of $f(x)=25-x^2$ from $x=0$ to $x=5$ using five approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?
(b) Repeat part (a) using left endpoints.

Kevin Shryock
Kevin Shryock
Numerade Educator
09:00

Problem 5

(a) Estimate the area under the graph of $f(x)=1+x^2$ from $x=-1$ to $x=2$ using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles.
(b) Repeat part (a) using left endpoints.
(c) Repeat part (a) using midpoints.
(d) From your sketches in parts (a)(c), which appears to be the best estimate?

Frank Lin
Frank Lin
Numerade Educator
07:55

Problem 6

(a) Graph the function $f(x)=e^{-x^2},-2 \leqslant x \leqslant 2$.
(b) Estimate the area under the graph of $f$ using four approximating rectangles and taking the sample points to be
(i) right endpoints
(ii) midpoints
In each case sketch the curve and the rectangles.
(c) Improve your estimates in part (b) by using eight rectangles.

Sam Low
Sam Low
Numerade Educator
01:14

Problem 7

With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of $n$, using looping. (On a TI use the Is> command or a For-EndFor loop, on a Casio use Isz, on an HP or in BASIC use a FOR-NEXT loop.) Compute the sum of the areas of approximating rectangles using equal subintervals and right endpoints for $n=10,30$, and 50 . Then guess the value of the exact area.
The region under $y=\sin x$ from 0 to $\pi$

James Kiss
James Kiss
Numerade Educator
01:14

Problem 8

With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of $n$, using looping. (On a TI use the Is> command or a For-EndFor loop, on a Casio use Isz, on an HP or in BASIC use a FOR-NEXT loop.) Compute the sum of the areas of approximating rectangles using equal subintervals and right endpoints for $n=10,30$, and 50 . Then guess the value of the exact area.
The region under $y=1 / x^2$ from 1 to 2

James Kiss
James Kiss
Numerade Educator
01:54

Problem 9

Some computer algebra systems have commands that will draw approximating rectangles and evaluate the sums of their areas, at least if $x_{\%}^*$ is a left or right endpoint. (For instance, in Maple use leftbox, rightbox, leftsum, and rightsum.)
(a) If $f(x)=\sqrt{x}, 1 \leqslant x \leqslant 4$, find the left and right sums for $n=10,30$, and 50 .
(b) Illustrate by graphing the rectangles in part (a).
(c) Show that the exact area under $f$ lies between 4.6 and 4.7 .

James Kiss
James Kiss
Numerade Educator
02:25

Problem 10

(a) If $f(x)=\sin (\sin x), 0 \leqslant x \leqslant \pi / 2$, use the commands discussed in Exercise 9 to find the left and right sums for $n=10,30$, and 50 .
(b) Illustrate by graphing the rectangles in part (a).
(c) Show that the exact area under $f$ lies between 0.87 and 0.91 .

Sam Low
Sam Low
Numerade Educator
01:56

Problem 11

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds.
$$
\begin{array}{|l|l|r|r|r|r|r|r|}
\hline t(\mathrm{~s}) & 0 & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 \\
\hline v(\mathrm{ft} / \mathrm{s}) & 0 & 6.2 & 10.8 & 14.9 & 18.1 & 19.4 & 20.2 \\
\hline
\end{array}
$$

Lucas Finney
Lucas Finney
Numerade Educator
01:09

Problem 12

Speedometer readings for a motorcycle at 12 -second intervals are given in the table.
(a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals.
(b) Give another estimate using the velocities at the end of the time periods.
(c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain.
$$
\begin{array}{|l|c|c|c|c|c|c|}
\hline t(\mathrm{~s}) & 0 & 12 & 24 & 36 & 48 & 60 \\
\hline v(\mathrm{ft} / \mathrm{s}) & 30 & 28 & 25 & 22 & 24 & 27 \\
\hline
\end{array}
$$

Fuzail Shakir
Fuzail Shakir
Numerade Educator
00:59

Problem 13

Oil leaked from a tank at a rate of $r(t)$ liters per hour. The rate decreased as time passed and values of the rate at 2-hour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.
$$
\begin{array}{|l|c|c|c|c|c|c|}
\hline t(\mathrm{~h}) & 0 & 2 & 4 & 6 & 8 & 10 \\
\hline r(t)(\mathrm{L} / \mathrm{h}) & 8.7 & 7.6 & 6.8 & 6.2 & 5.7 & 5.3 \\
\hline
\end{array}
$$

James Kiss
James Kiss
Numerade Educator
01:52

Problem 14

When we estimate distances from velocity data, it is sometimes necessary to use times $t_0, t_1, t_2, t_3, \ldots$ that are not equally spaced. We can still estimate distances using the time periods $\Delta t_i=t_i-t_{i-1}$. For example, on May 7,1992 , the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table, provided by NASA, gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters.
$$
\begin{array}{|l|c|c|}
\hline \text { Event } & \text { Time (s) } & \text { Velocity (ft/s) } \\
\hline \text { Launch } & 0 & 0 \\
\text { Begin roll maneuver } & 10 & 185 \\
\text { End roll maneuver } & 15 & 319 \\
\text { Throttle to 89\% } & 20 & 447 \\
\text { Throttle to 67\% } & 32 & 742 \\
\text { Throttle to 104\% } & 59 & 1325 \\
\text { Maximum dynamic pressure } & 62 & 1445 \\
\text { Solid rocket booster separation } & 125 & 4151 \\
\hline
\end{array}
$$
Use these data to estimate the height above Earth's surface of the space shuttle Endeavour, 62 seconds after liftoff.

James Kiss
James Kiss
Numerade Educator
03:19

Problem 15

The velocity graph of a braking car is shown. Use it to estimate the distance traveled by the car while the brakes are applied.
GRAPH CANT COPY

Lucas Finney
Lucas Finney
Numerade Educator
03:06

Problem 16

The velocity graph of a car accelerating from rest to a speed of $120 \mathrm{~km} / \mathrm{h}$ over a period of 30 seconds is shown. Estimate the distance traveled during this period.
GRAPH CANT COPY

Lucas Finney
Lucas Finney
Numerade Educator
02:01

Problem 17

Use Definition 2 to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit.
$f(x)=\sqrt[4]{x}, \quad 1 \leqslant x \leqslant 16$

Sam Low
Sam Low
Numerade Educator
01:51

Problem 18

Use Definition 2 to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit.
$f(x)=\frac{\ln x}{x}, \quad 3 \leqslant x \leqslant 10$

Sam Low
Sam Low
Numerade Educator
02:15

Problem 19

Use Definition 2 to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit.
$f(x)=x \cos x, \quad 0 \leqslant x \leqslant \pi / 2$

Lucas Finney
Lucas Finney
Numerade Educator
02:31

Problem 20

Determine a region whose area is equal to the given limit. Do not evaluate the limit.
$\lim _{n \rightarrow \infty} \sum_{i=1}^n \frac{2}{n}\left(5+\frac{2 i}{n}\right)^{10}$

Lucas Finney
Lucas Finney
Numerade Educator
02:19

Problem 21

Determine a region whose area is equal to the given limit. Do not evaluate the limit.
$\lim _{s \rightarrow \infty} \sum_{j=1}^n \frac{\pi}{4 n} \tan \frac{i \pi}{4 n}$

Carson Merrill
Carson Merrill
Numerade Educator
01:51

Problem 22

(a) Use Definition 2 to find an expression for the area under the curve $y=x^3$ from 0 to 1 as a limit.
(b) The following formula for the sum of the cubes of the first $n$ integers is proved in Appendix E. Use it to evaluate the limit in part (a).
$$
1^3+2^3+3^3+\cdots+n^3=\left[\frac{n(n+1)}{2}\right]^2
$$

Frank Lin
Frank Lin
Numerade Educator
View

Problem 23

(a) Express the area under the curve $y=x^5$ from 0 to 2 as a limit.
(b) Use a computer algebra system to find the sum in your expression from part (a).
(c) Evaluate the limit in part (a).

Gio Maya
Gio Maya
Numerade Educator
01:32

Problem 24

Find the exact area of the region under the graph of $y=e^{-x}$ from 0 to 2 by using a computer algebra system to evaluate the sum and then the limit in Example 3(a). Compare your answer with the estimate obtained in Example 3(b).

Frank Lin
Frank Lin
Numerade Educator
05:43

Problem 25

Find the exact area under the cosine curve $y=\cos x$ from $x=0$ to $x=b$, where $0 \leqslant b \leqslant \pi / 2$. (Use a computer algebra system both to evaluate the sum and compute the limit.) In particular, what is the area if $b=\pi / 2$ ?

Robert Daugherty
Robert Daugherty
Numerade Educator
01:46

Problem 26

(a) Let $A_n$ be the area of a polygon with $n$ equal sides inscribed in a circle with radius $r$. By dividing the polygon into $n$ congruent triangles with central angle $2 \pi / n$, show that $A_n=\frac{1}{2} n r^2 \sin (2 \pi / n)$.
(b) Show that $\lim _{n \rightarrow \infty} A_n=\pi r^2$.

James Kiss
James Kiss
Numerade Educator