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Chapter 5

INTEGRALS

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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 ten rectangles in each case.

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Problem 2

Use six rectangles to find estimates of each type for the
area under the given graph of $f$ from $x=0$ to $x=12$ .
$$
\begin{array}{ll}{\text { (i) } L_{6}} & {\text { (sample points are left endpoints) }} \\ {\text { (ii) } R_{6}} & {\text { (sample points are right endpoints) }} \\ {\text { (iii) } M_{6}} & {\text { (sample points are midpoints) }}\end{array}
$$
(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 esti-
mate? Explain.

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Problem 3

(a) Estimate the area under the graph of $f(x)=\cos x$ from
$x=0$ to $x=\pi / 2$ 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.

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Problem 4

(a) Estimate the area under the graph of $f(x)=\sqrt{x}$ from
$x=0$ to $x=4$ 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.

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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?

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Problem 6

(a) Graph the function
$f(x)=x-2 \ln x \quad 1 \leqslant x \leqslant 5$
(b) Estimate the area under the graph of $f$ using four
approximating rectangles and taking the sample points
to be (i) right endpoints and (ii) midpoints. In each
case sketch the curve and the rectangles.
(c) Improve your estimates in part (b) by using eight
rectangles.

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Problem 7

Evaluate the upper and lower sums for $f(x)=2+\sin x$
$0 \leqslant x \leqslant \pi,$ with $n=2,4,$ and $8 .$ Illustrate with diagrams
like Figure $14 .$

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Problem 8

Evaluate the upper and lower sums for $f(x)=1+x^{2}$
$-1 \leqslant x \leqslant 1,$ with $n=3$ and $4 .$ Illustrate with diagrams
like Figure $14 .$

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Problem 9

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.
$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline t(s) & {0} & {0.5} & {1.0} & {1.5} & {2.0} & {2.5} & {3.0} \\ \hline v(f t / s) & {0} & {6.2} & {10.8} & {14.9} & {18.1} & {19.4} & {20.2} \\ \hline\end{array}$$

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Problem 10

Speedometer readings for a motorcycle at 12 -second intervals are given in the table.
$$\begin{array}{|c|c|c|c|c|c|c|}\hline t(s) & {0} & {12} & {24} & {36} & {48} & {60} \\ \hline v(f t / s) & {30} & {28} & {25} & {22} & {24} & {27} \\ \hline\end{array}$$
(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.

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Problem 11

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 two hour time intervals are shown in the table. Find lower and
upper estimates for the total amount of oil that leaked out.
$$\begin{array}{|c|c|c|c|c|c|}\hline t(h) & {0} & {2} & {4} & {6} & {8} & {10} \\ \hline r(t) & {(L / h)} & {8.7} & {7.6} & {6.8} & {6.2} & {5.7} & {5.3} \\ \hline\end{array}$$

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Problem 12

When we estimate distances from velocity data, it is some- times 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. Use these data to estimate the height above the earth's surface of the space shuttle Endeavour, 62 seconds after liftoff.

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Problem 13

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

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Problem 14

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.

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Problem 15

$15-16=$ Use Definition 2 to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit.
$$f(x)=\frac{2 x}{x^{2}+1}, \quad 1 \leqslant x \leqslant 3$$

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Problem 16

$15-16=$ Use Definition 2 to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit.
$$f(x)=x^{2}+\sqrt{1+2 x}, \quad 4 \leqslant x \leqslant 7$$

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Problem 17

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

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Problem 18

(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 B. Use it to evaluate the limit in part (a).
$$1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\left[\frac{n(n+1)}{2}\right]^{2}$$

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Problem 19

Let $A$ be the area under the graph of an increasing continuous function $f$ from $a$ to $b,$ and let $L_{n}$ and $R_{n}$ be the approximations to $A$ with $n$ sub intervals using left and right
endpoints, respectively.
(a) How are $A, L_{n},$ and $R_{n}$ related?
(b) Show that
$$R_{n}-L_{n}=\frac{b-a}{n}[f(b)-f(a)]$$
Then draw a diagram to illustrate this equation by showing that the $n$ rectangles representing $R_{n}-L_{n}$ can be reassembled to form a single rectangle whose area is the right side of the equation.
(c) Deduce that
$$R_{n}-A<\frac{b-a}{n}[f(b)-f(a)]$$

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Problem 20

If $A$ is the area under the curve $y=e^{x}$ from 1 to $3,$ use Exercise 19 to find a value of $n$ such that $R_{n}-A<0.0001$

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Problem 21

(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).

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Problem 22

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$(b)$

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Problem 23

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 $b=\pi / 2 ?$

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Problem 24

(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
$$A_{n}=\frac{1}{2} n r^{2} \sin \left(\frac{2 \pi}{n}\right)$$
(b) Show that $\lim _{n \rightarrow \infty} A_{n}=\pi r^{2} .[$Hint$:$ Use Equation 1.4 .6 on page $42 . ]$

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