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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 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

(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) $\mathrm{L}_{6}$ (sample points are left endpoints)
(ii) $\mathrm{R}_{6}$ (sample points are right endpoints)
(iii) $\mathrm{M}_{6}$ (sample points are midpoints)
(b) Is $\mathrm{L}_{6}$ an underestimate or overestimate of the true area?
(c) Is $\mathrm{R}_{6}$ an underestimate or overestimate of the true area?
(d) Which of the numbers $\mathrm{L}_{6}, \mathrm{R}_{6}$ , or $\mathrm{M}_{6}$ gives the best estimate? 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 $\mathrm{f}(\mathrm{x})=\sqrt{\mathrm{x}}$ from $\mathrm{x}=0$ to $\mathrm{x}=4$ 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 ( $2 )-(c),$ which appears to $\quad$ be the best estimate?

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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 approx- imating 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 8 rectangles.

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Problem 7

$7-8$ 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 $n=10,30,50,$ and 100 . Then guess the value of the exact area.

The region under $y=x^{4}$ from 0 to 1

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Problem 8

$7-8$ 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 $n=10,30,50,$ and 100 . Then guess the value of the exact area.

The region under $\mathrm{y}=\cos x$ from 0 to $\pi / 2$

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Problem 9

Some computer algebra systems have commands that will draw approximating rectangles and evaluate the sums of their areas, at least if $x_{1}^{\star}$ is a left or right endpoint. (For instance, in Maple use leftbox, rightbox, leftsum, and right-sum.)
(a) If $f(x)=1 /\left(x^{2}+1\right), 0 \leqslant x \leqslant 1,$ 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 0.780 and $0.791 .$

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Problem 10

(a) If $f(x)=\ln x, 1 \leqslant x \leqslant 4,$ use the commands discussed in Exercise 9 to find the left and right sums for $n=10$ , $\quad 30,$ and $50 .$
(b) Illustrate by graphing the rectangles in part (a)
(c) Show that the exact area under $f$ lies between 2.50 and $2.59 .$

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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.

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

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Problem 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.
$$\begin{array}{|c|c|c|c|c|c|c|}\hline t(s) & {0} & {12} & {24} & {36} & {48} & {60} \\ \hline v(f t / s) & {30} & {28} & {25} & {22} & {24} & {27} \\ \hline\end{array}$$

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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 two-hour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.

$$\begin{array}{|c|c|c|c|c|c|c|}\hline t(h) & {0} & {2} & {4} & {6} & {8} & {10} \\ \hline r(t) & {(L / h)} & {8.7} & {7.6} & {6.8} & {6.2} & {5.7} & {5.3} \\ \hline\end{array}$$

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Problem 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 $\Delta t_{1}=t_{1}-t_{1-1}$ . 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 Endeavour, 62 seconds after liftoff.

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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.

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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.

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Problem 17

$17-19$ Use Definition 2 to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit.

$f(x)=\sqrt[4]{x}, 1 \leqslant x \leqslant 16$

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Problem 18

17-19 Use Definition 2 to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit.

$f(x)=\frac{\ln x}{x}, \quad 3 \leq x \leqslant 10$

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Problem 19

$17-19$ Use Definition 2 to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit.

$f(x)=x \cos x, \quad 0 \leqslant x \leqslant \pi / 2$

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Problem 20

$20-2 !$ Determine a region whose area is equal to the given limit. Do not evaluate the limit.

$$\lim _{n \rightarrow \infty} \sum_{l=1}^{n} \frac{2}{n}\left(5+\frac{2 i}{n}\right)^{10}$$

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Problem 21

$20-21$ Determine a region whose area is equal to the given limit. Do not evaluate the limit.

$$\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{4 n} \tan \frac{\mathrm{i} \pi}{4 \mathrm{n}}$$

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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).
$$1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\left[\frac{n(n+1)}{2}\right]^{2}$$

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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).

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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$(b)$

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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 $b=\pi / 2 ?$

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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
$$\mathrm{A}_{\mathrm{n}}=\frac{1}{2} \mathrm{nr}^{2} \sin \left(\frac{2 \pi}{\mathrm{n}}\right)$$
(b) Show that lim $_{\mathrm{n} \rightarrow \infty} \mathrm{A}_{\mathrm{n}}=\pi \mathrm{r}^{2} .[$ Hint: Use Equation 3.3 $.2 .]$

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