Deck 5: Applications of Integration

Question 1

Evaluate the integral.

Answer 1

Question 2

Find the general indefinite integral. $\int \frac{\sin 120 t}{\sin 60 t} d t$

A) $-\frac{\sin 120 t}{120}+C$
B) $-\frac{\cos 120 t}{120}+C$
C) $\frac{\sin 60 t}{30}+C$
D) $\frac{\sin 120 t}{120}+C$
E) $\frac{\cos 120 t}{30}+C$

Answer 2

$\frac{\sin 60 t}{30}+C$

Question 3

Find the area of the region that lies under the given curve. Round the answer to three decimal places. $y=\sqrt{3 x+2}, \quad 0 \leq x \leq 1$

A) $1.906$
B) $2.356$
C) $1.856$
D) $2.006$
E) $2.156$

Answer 3

$1.856$

Question 4

Evaluate the integral by making the given substitution.

Question 5

Evaluate the integral.

Question 6

Find the area of the region to three decimal places that lies under the given curve. $y=\sqrt{2 x+2}, \quad\quad0 \leq x \leq 1$

A) $1.724$
B) $1.874$
C) $4.974$
D) $1.769$
E) $2.729$

Question 7

Evaluate the integral. $\int_{0}^{1 / 2} \frac{4 d r}{\sqrt{1-r^{2}}}$

A) $\frac{\pi}{4}$
B) $\frac{1}{5}$
C) $\frac{\pi}{5}$
D) $\frac{1}{4}$
E) $\frac{2 \pi}{3}$

Question 8

Evaluate the indefinite integral. $\int \frac{e^{x}}{e^{x}+5} d x$

A) $-\frac{1}{2} \ln \left(e^{x}+5\right)+C$
B) $\ln \left(e^{x}+5\right)+C$
C) $\frac{1}{2} \ln \left(e^{x}+5\right)+C$
D) $-\ln \left(e^{x}+5\right)+C$
E) $\ln \left(e^{x}-5\right)+C$

Question 9

Evaluate the indefinite integral. $\int \frac{6+6 x}{\sqrt{7+6 x+3 x^{2}}} d x$

A) $-\sqrt{7+6 x+3 x^{2}}+C$
B) $-2 \sqrt{7+4 x+5 x^{2}}+C$
C) $2 \sqrt{7+6 x+3 x^{2}}+C$
D) $-5 \sqrt{7+6 x+3 x^{2}}+C$
E) $\sqrt{7+6 x+3 x^{2}}+C$

Question 10

Evaluate the integral.

Question 11

Evaluate the integral by making the given substitution. $\int x^{2} \sqrt{x^{3}+2} d x, \quad \quad u=x^{3}+2$

A) $-\frac{2}{9}\left(x^{3}+2\right)^{3 / 2}+C$
B) $\frac{2}{9}\left(x^{3}+2\right)^{3 / 2}$
C) $\frac{1}{9}\left(x^{3}+2\right)^{1 / 2}+C$
D) $\frac{2}{9}\left(x^{3}+2\right)^{1 / 2}+C$
E) $\frac{2}{9}\left(x^{3}+2\right)^{3 / 2}+C$

Question 12

Evaluate the integral.

Question 13

Evaluate the definite integral.

Question 14

Evaluate the integral if it exists. $\int_{0}^{1} 6 x^{2} \cos \left(x^{3}\right) d x$

A) 0
B) $\cos (3)$
C) $\sin (6)$
D) $3 \sin (3)$
E) none of these

Question 15

Evaluate the integral if it exists. $\int_{0}^{1} 3 x^{2} \cos \left(x^{2}\right) d x$

A) 0
B) $2 \sin (2)$
C) $\sin (3)$
D) $\cos (2)$
E) none of these

Question 16

Evaluate the definite integral.

Question 17

Evaluate the indefinite integral.

Question 18

Evaluate the indefinite integral. $\int \cos ^{6} x \sin x d x$

A) $-\frac{1}{7} \sin ^{7} x+C$
B) $\frac{1}{7} \sin ^{7} x+C$
C) $\frac{1}{7} \cos ^{7} x+C$
D) $-\frac{1}{7} \cos ^{7} x+C$
E) $\frac{1}{7} \cos ^{7} x+C$ .

Question 19

Evaluate the indefinite integral.

Question 20

Evaluate the indefinite integral. $\int 7 e^{\cos x} \sin x d x$

A) $-e^{\cos x} \sin x+C$
B) $-7 e^{\cos x}+C$
C) $e^{7 \sin x}+C$
D) $7 e^{\cos x} \sin x+C$
E) $-7 \sin \left(e^{\cos x}\right)+C$

Question 21

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h(x)=\int_{1}^{\sqrt{x}} \frac{z^{2}}{z^{4}+1} d z$

A) $\frac{\sqrt{x+1}}{x^{2}+2}$
B) $\frac{\sqrt{x^{2}+1}}{2}$
C) none of these
D) $\frac{\sqrt{x}}{x^{2}+1}$
E) $\frac{\sqrt{x}}{2\left(x^{2}+1\right)}$

Question 22

The velocity function (in meters per second) is given for a particle moving along a line. Find the distance traveled by the particle during the given time interval. $v(t)=8 t-8,0 \leq t \leq 5$

A) 36 m
B) 72 m
C) 100 m
D) 64 m
E) 68 m

Question 23

Find the area of the region that lies to the right of the y-axis and to the left of the parabola .

Question 24

An animal population is increasing at a rate of per year (where t is measured in years). By how much does the animal population increase between the fourth and tenth years?

Question 25

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

Question 26

An animal population is increasing at a rate of $16+51 t$ per year (where t is measured in years). By how much does the animal population increase between the fourth and tenth years?

A) $2248$
B) $2288$
C) $2338$
D) $2258$
E) $2238$

Question 27

Evaluate $\frac{d}{d x} \int_{0}^{x}\left(e^{\arccos t}\right) d t$

A) $e^{\arccos x}$
B) $e^{\arccos t}$
C) $x$
D) $e^{x}-\arccos x$
E) $e^{\operatorname{arccost} \frac{x}{4}}$

Question 28

Estimate the area from 0 to 5 under the graph of $f(x)=25-x^{2}$ using five approximating rectangles and right endpoints.

A) $65$
B) $60$
C) $170$
D) $270$
E) $70$

Question 29

The velocity of a car was read from its speedometer at ten-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car. $t(\mathrm{s})$ $v(\mathrm{mi} / \mathrm{h})$ $t(\mathrm{s})$ $v(\mathrm{mi} / \mathrm{h})$ 0 0
60 54 10 $34$ 70 $67$ 20 $43$ 80 $77$ 30 $36$ 90 $37$ 40 $45$ 100 $42$ 50 $45$

A) 1.2 miles
B) 1.8 miles
C) 0.8 miles
D) 2.4 miles
E) 0.6 miles

Question 30

The marginal cost of manufacturing x yards of a certain fabric is $C^{\prime}(x)=3-0.01 x+0.000006 x^{2}$ in dollars per yard. Find the increase in cost if the production level is raised from $2500$ yards to $6500$ yards.

A) $\$ 278,000.00$
B) $\$ 248,000.00$
C) $\$ 268,000.00$
D) $\$ 288,000.00$
E) $\$ 258,000.00$

Question 31

Evaluate the integral. $\int_{1}^{4} \frac{x^{2}+6}{\sqrt{x}} d x$

A) $24.4$
B) $34.4$
C) $39.4$
D) $29.4$
E) $44.4$

Question 32

The area of the region that lies to the right of the y-axis and to the left of the parabola $x=6 y-y^{2}$ (the shaded region in the figure) is given by the integral $\int_{0}^{5}\left(6 y-y^{2}\right) d y$ . Find the area..

A) $17.5$
B) $12.5$
C) $10.25$
D) $\frac{275}{6}$
E) $\frac{25}{6}$

Question 33

If $h^{\prime}$ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 2 and 5 ?

A) $\int_{2}^{5} h^{t}(t) d t$
B) $h^{\prime}(5)-h^{\prime}(2)$
C) $\int_{5}^{2} h(t) d t$

Question 34

If $F(x)=\int_{1}^{x} f(t) ~d t$ where $f(t)=\int_{1}^{t^{2}} \frac{\sqrt{2+u^{2}}}{u}$ find $F^{\prime \prime}(2)$ .

A) $3 \sqrt{2}$
B) $6 \sqrt{2}$
C) $6 \sqrt{3}$
D) $\frac{3 \sqrt{2}}{2}$
E) $3 \sqrt{3}$

Question 35

Evaluate the integral if it exists. $\int\left(\frac{5-x}{x}\right)^{2} d x$

A) $1-10 \ln x+C$
B) $x-10 \ln x-\frac{25}{x}+C$
C) $\ln x-25 x+C$
D) $x-\frac{1}{10 \ln x}+C$
E) none of these

Question 36

Evaluate the integral. $\int_{a / 8}^{s / 6} \sin ~t ~d t$

A) $-0.943$
B) $0.857$
C) $1.057$
D) $0.057$
E) $0.557$

Question 37

The acceleration function (in m/ $S^{2}$ ) and the initial velocity are given for a particle moving along a line. Find the velocity at time t and the distance traveled during the given time interval. $a(t)=t+4, v(0)=5,0 \leq t \leq 10$

A) $v(t)=\frac{t^{2}}{2}+5 t \mathrm{~m} / \mathrm{s}, 566 \frac{2}{3} \mathrm{~m}$
B) $v(t)=\frac{t^{2}}{2}+5 \mathrm{~m} / \mathrm{s}, 616 \frac{2}{3} \mathrm{~m}$
C) $v(t)=\frac{t^{2}}{2}+5 t \mathrm{~m} / \mathrm{s}, 591 \frac{2}{3} \mathrm{~m}$
D) $v(t)=\frac{t^{2}}{2}+4 t+5 \mathrm{~m} / \mathrm{s}, 416 \frac{2}{3} \mathrm{~m}$
E) $v(t)=\frac{t^{2}}{2}+4 t+5 \mathrm{~m} / \mathrm{s}, 516 \frac{2}{3} \mathrm{~m}$

Question 38

Evaluate the integral. $\int_{0}^{9}\left(6+6 y-y^{2}\right) d y$

A) 54
B) $34$
C) $94$
D) $84$
E) $74$

Question 39

Evaluate the definite integral. $\int_{0}^{\pi / 8} \sin~ 5 t ~d t$

A) $0.28$
B) $3.28$
C) $1.28$
D) $2.28$
E) $1.08$

Question 40

Evaluate the integral if it exists.

Question 41

Evaluate by interpreting it in terms of areas.

Question 42

Express the integral as a limit of sums. Then evaluate the limit.

Question 43

Use the Midpoint Rule with n = 10 to approximate the integral. $\int_{1}^{2} \sqrt{4+t^{2}} ~~d t$

A) $7.510716$
B) $1.510716$
C) $12.510716$
D) $2.510716$
E) $10.510716$

Question 44

Use the given graph of $f$ to find the Riemann sum with six subintervals. Take the sample points to be left endpoints.

A) 8
B) 6
C) 4
D) 3.5
E) 4.5

Question 45

Use the Midpoint Rule with n = 5 to approximate the integral. $\int_{0}^{10} 5 \sin \sqrt{q}~~ d q$ Round your answer to three decimal places.

A) $36.909$
B) $36.409$
C) $31.409$
D) $35.909$
E) $37.709$

Question 46

Find the area of the region that lies under the given curve.

Question 47

Find an expression for the area under the graph of as a limit. Do not evaluate the limit.

Question 48

Evaluate by interpreting it in terms of areas.

Question 49

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

Question 50

Find the derivative of the function.

Question 51

Given that , find .

Question 52

Evaluate the integral. $\int_{-2}^{5}\left|7 x-x^{2}\right| d x$ Round your answer to the nearest hundredth.

A) $232.5$
B) $64.5$
C) $112.5$
D) $62.5$
E) $212.5$

Question 53

Evaluate the Riemann sum for $f(r)=36-r^{2}, 0 \leq r \leq 2$ with four subintervals, taking the sample points to be right endpoints.

A) $68.25$
B) $69.75$
C) $70.75$
D) $70.25$
E) $69.25$

Question 54

Express the limit as a definite integral on the given interval.

Question 55

Evaluate by interpreting it in terms of areas.

Question 56

If $\int_{0}^{6} f(x) d x=15$ and $\int_{0}^{4} f(x) d x=6$ , find $\int_{4}^{6} f(x) d x$ .

A) 6
B) 21
C) 15
D) 9
E) $-15$

Question 57

A table of values of an increasing function is shown. Use the table to find an upper estimate of . -45 -37 -27 9 10 23

Question 58

Find the derivative of the function.

Question 59

Find a function such that for x > 0 and some number
a.

Question 60

If , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints.

Question 61

Use the definition of area to find the area of the region under the graph of f on [a, b] using the indicated choice of c_k. = + 6x + 1, [ , 1], c_k is the right endpoint

Question 62

Find an expression for the area under the graph of as a limit. Do not evaluate the limit.

Question 63

The velocity graph of a car accelerating from rest to a speed of 7 km/h over a period of 10 seconds is shown. Estimate to the nearest integer the distance traveled during this period. Use a right sum with .

Question 64

The velocity graph of a braking car is shown. Use it to estimate to the nearest foot the distance traveled by the car while the brakes are applied.Use a left sum with n = 7.

Question 65

Approximate the area under the curve from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

Question 66

Approximate the area under the curve $y=\sin x$ from $0 \text { to } \frac{\pi}{2}$ using ten approximating rectangles of equal widths and right endpoints. The choices are rounded to the nearest hundredth.

A) 0.36
B) 0.02
C) 0.72
D) 0.98
E) 1.08

Question 67

Determine a region whose area is equal to $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{2 n} \tan \frac{i \pi}{2 n}$ .

A) $y=\tan x, 0 \leq x \leq \frac{\pi}{3}$
B) $y=\tan x, 0 \leq x \leq \frac{\pi}{5}$
C) $y=\tan x, 0 \leq x \leq \frac{\pi}{6}$
D) $y=\tan x, 0 \leq x \leq \frac{\pi}{4}$
E) $y=\tan x, 0 \leq x \leq \frac{\pi}{2}$

Question 68

By reading values from the given graph of , use five rectangles to find a lower estimate, to the nearest whole number, for the area from 0 to 10 under the given graph of .

Question 69

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 a lower estimate for the distance that she traveled during these three seconds. 2.8
3.5
6.9
8.2
12.2
16.3

Question 70

Use the definition of area to find the area of the region under the graph of f on [a, b] using the indicated choice of c_k. = , [0, 2], c_k is the left endpoint