## Review Exercises for Applied Calculus Topic: The Integral

Summary for This Topic
True/False Quiz
Index of Review Exercises
Everything for Calculus
Utility: Numerical Integration Utility
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Note Use proper graphing calculator format to input your answers (spaces are ignored). Here are some examples of correctly formatted expressions involving logarithms and exponentials.

Evaluate each of the following integrals.

I 1.
 (x3 - 6x + 4) dx
=
I 2.
 6x + 6x2 dx
=
I 3.
 1u2 - 4u du
=
I 4.
 (2ex + 3x) dx
=
I 5.
 (u + 2)5 dx
=
I 6.
 (2x - 3)4 dx
=
I 7.
 x2(x3 + 1)4 dx
=
I 8.
 xx2 + 1 dx
=
I 9.
 5e-2t dt
=
I 10.
 2.4x0.2 x1.2 - 4 dx
=
I 11.
 2x + 1(2x2 + 2x - 3)2 dx
=
I 12.
 0.6e0.3t1+2e0.3t dt
=
I 13.
 (3x+2)e(3x2+4x) dx
=
I 14.
 12(x - 2)(3x2 - 12x + 5)3 dx
=
DI 1.
 1  0 (x3 + 2x) dx
=
DI 2.
 1  -1 (x3+x2) dx
=
DI 3.
 2  1 3x2 dx
=
DI 4.
 1  0 (e-x + x) dx
=
DI 5.
 2  0 x2(x3 + 1)1/2 dx
=
DI 6.
 1  -1 (xe-0.3x2) dx
=
DI 7.
 1 0 7(7y + 1)1/3 dy
=
DI 8.
 1 0 y(1+y)3 dy
=

In each of the following, find the area of the region described.

A 1. Bounded by y = 1-x2, the x-axis, and the lines vv = -1 and vv = 1

 Area =

A 2. Bounded by y = 1-x4, the x-axis, and the lines vv = -1 and vv = 1

 Area =

A 3. Bounded by y = x4 - x2 and the x-axis

 Area =

A 4. Bounded by y = 1/x, the x-axis, and the lines vv = 1 and vv = e2

 Area =

A 5. Bounded by y = ex - e-x, the x-axis, and the lines vv = -1 and vv = 1

 Area =

Evaluate the left Riemann sum approximation for each of the following integrals. Round your answers to two decimal places.

 R 1. 1 -1 e-x2 dx;     n = 4
 Riemann sum =

 R 2. 2 0.5 ln x dx;    n = 3
 Riemann sum =

 R 3. 1 0 (1 + x1/2)1/2 dx;    n = 2
 Riemann sum =

 R 4. 1 0 4 dx1 + x2 ; n = 5
 Riemann sum =

The next two require the use of technology, such as the On-Line Numerical Integration Utility.

 R5. 1 0 4 dx1 + x2 n = 100
 Riemann sum =

 R6. 1 0 (1 + x1/2)1/2 dx;    n = 150
 Riemann sum =

App 1. Baby Buggies The marginal cost of Better Baby Buggies Inc.'s Turbo model is

 50 + 402x+1

for the xth buggy made in a week, and the company's fixed costs amount to \$50,000 each week. Find the total cost to make x Turbo buggies in one week.

 Answer: Cost of x Turbo buggies = \$

App 2. Wire Transactions The accompanying graph shows the number of transactions handled by Western Union for the years 1987-1994.*

* Source: Company Reports/The New York Times, September 24, 1994.

These data can be approximated by

Q(t) = 9.284e0.117t     Million transactions per year

where t is time in years since 1987. Regarding the modeal as a measure of the rate of change of the total number of transactions, estimate the total number of Western Union transactions from 1985 (t = -2) to 1995 (t = 8). Round your answer to the nearest one million transactions.

 Answer: A total of million transactions.

App 3. Magazine Sales The following chart shows annual newsstand sales of all magazines in the U.S. (in millions of magazines)*

*Source: Vos, Gruppo & Capell Inc./The New York Times Dec. 8, 1993, p. D6.

(a) Model the annual sales with a linear function s(t) using the data from 1982 (t = 0) and 1992 (t = 10).

 Answer: s(t) = million magazines per year

(b) Use your model to estimate the number of magazines sold in the U.S. from the start of 1980 (t = -2) to the start of 1990.

 Answer: Total sales were approximately million magazines.

App 4. Cell Phones The rate of production of cellular phones at your electronics plant is given by the following Cobb-Douglas production function:

p = 10x0.3y0.7     telephones per year

where x is the number of employees, and y is the daily operating budget (in dollars). You have decided to increase the daily operating budget linearly from the present level of \$6,000 to \$10,000 over the next two years. Your plant employs 100 workers. Estimate the total production of cellular phones over the next two years. Round your answer to the nearest 100 cellular phones. 42,900

App 5. Growth Hormone Sales Annual sales of Eli Lilly Corp.'s human growth hormone Humatrope can be modeled by
 S(t) = 0.40e2.7t160 + 0.0025e2.7t million dollars per year

where t represents the number of years since the drug's approval by the FDA in 1987.* (Take t = 0 to correspond to January 1, 1987.)
(a) Use the model to estimate the value of total sales of Humatrope from January, 1987 through December, 1997. Round your answer to the nearest \$10 million.

 Answer: Total sales \$ million magazines per year

(b) Calculate both S'(6) and 06S(t) dt (round both answers to three significant digits) and carefully explain what is measured by each.

* The model is a very crude one, based on 1991 sales data, total sales data through May, 1992 and very rough estimates of the potential market and selling price. Source: Senate Judiciary Committee; Subcommittee on Antitrust and Monopoly/The New York Times, May 14, 1992, p. D1.