In Exercises 6168, use a graphing utility to graph the function over the interval. Find

In Exercises 18, sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. 2 dx 2 0 3 dx

In Exercises 18, sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. 4 0 2 dx

In Exercises 18, sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. 2x 1 dx 5 0 x 1 dx 4

In Exercises 18, sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. 3 0 2x 1 dx 5

In Exercises 18, sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral.3 2 x  1 dx

In Exercises 18, sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. 4 1  x  2 dx

In Exercises 18, sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. dx 3 3 9  x2 dx

In Exercises 18, sketch the region whose area is represented by the definite integral. Then use a geometric formula to evaluate the integral. 2 0 4  x2 dx

In Exercises 9 and 10, use the values and to evaluate the definite integral. fx  gx dx 5 0  fx gx dx 5 0 fx  3gx dx 5 0 4 fx dx 5

In Exercises 9 and 10, use the values and to evaluate the definite integral. fx dx 5 0 2gx dx fx  fx dx 5 5 fx dx 0

In Exercises 1118, find the area of the region. y
x  x2

In Exercises 1118, find the area of the region. y
1  x4

In Exercises 1118, find the area of the region. y
1 x2

In Exercises 1118, find the area of the region. y
2 x

In Exercises 1118, find the area of the region. y
3ex
2

In Exercises 1118, find the area of the region. y
2ex
2

In Exercises 1118, find the area of the region. y
2ex
2

In Exercises 1118, find the area of the region.y
2ex
2

In Exercises 1942, evaluate the definite integral. 0 2x dx

In Exercises 1942, evaluate the definite integral. 7 2 3v dv

In Exercises 1942, evaluate the definite integral. 0 1 2x 1 dx 7 2 3v dv 1 0 2

In Exercises 1942, evaluate the definite integral. 5 2 3x 4 dx 0

In Exercises 1942, evaluate the definite integral. 1 2t  12 dt

In Exercises 1942, evaluate the definite integral.1 0 1  2x2 dx

In Exercises 1942, evaluate the definite integral.dx 3 0 x  23 dx

In Exercises 1942, evaluate the definite integral.2 2 x  34 dx 3

In Exercises 1942, evaluate the definite integral.dx 1 1  3 t  2 dt

In Exercises 1942, evaluate the definite integral.4 1 2 x dx

In Exercises 1942, evaluate the definite integral.4 1 2u  1 u du

In Exercises 1942, evaluate the definite integral.1 0 x  x 3 dx

In Exercises 1942, evaluate the definite integral.0 1 t1
3  t2
3 dt

In Exercises 1942, evaluate the definite integral.4 0 x1
2 x1
4 dx 0

In Exercises 1942, evaluate the definite integral.dx 4 0 1 2x 1 dx

In Exercises 1942, evaluate the definite integral.2 0 x 1 2x2 dx

In Exercises 1942, evaluate the definite integral.dx 1 0 e2x dx

In Exercises 1942, evaluate the definite integral.2 1 e1x dx

In Exercises 1942, evaluate the definite integral. dx 3 1 e3
x x2 dx

In Exercises 1942, evaluate the definite integral. 1 1 ex  ex dx 3

In Exercises 1942, evaluate the definite integral. 0 e2xe2x 1 dx

In Exercises 1942, evaluate the definite integral. 1 0 ex ex 1 dx

In Exercises 1942, evaluate the definite integral.dx 2 0 x 1 4x2 dx

In Exercises 1942, evaluate the definite integral. 1 0 e2x e2x 1 dx

In Exercises 4346, evaluate the definite integral by the most convenient method. Explain your approach. 1 1 4x dx

In Exercises 4346, evaluate the definite integral by the most convenient method. Explain your approach. 3 0 2x  3 dx

In Exercises 4346, evaluate the definite integral by the most convenient method. Explain your approach. dx 4 0 2  x  2 dx

In Exercises 4346, evaluate the definite integral by the most convenient method. Explain your approach.4 4 4  x dx 4

In Exercises 4750, evaluate the definite integral by hand. Then use a symbolic integration utility to evaluate the definite integral. Briefly explain any differences in your results. dx 2 1 x x2  9 dx

In Exercises 4750, evaluate the definite integral by hand. Then use a symbolic integration utility to evaluate the definite integral. Briefly explain any differences in your results. 3 2 x 1 x2 2x  3 dx 2

In Exercises 4750, evaluate the definite integral by hand. Then use a symbolic integration utility to evaluate the definite integral. Briefly explain any differences in your results. dx 3 0 2ex 2 ex dx

In Exercises 4750, evaluate the definite integral by hand. Then use a symbolic integration utility to evaluate the definite integral. Briefly explain any differences in your results.2 1 2 ln x3 x dx 3

In Exercises 5156, evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral. x 4 dx 3 1 4x  3 dx 2

In Exercises 5156, evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral. 2 0 x 4 dx 3

In Exercises 5156, evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral.x1  x dx 1 0 x  x3 dx

In Exercises 5156, evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral. 1 0 x1  x dx 1

In Exercises 5156, evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral. 4 2 3x2 x3  1 dx

In Exercises 5156, evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral.ln 6 0 ex 2 dx

In Exercises 5760, find the area of the region bounded by the graphs. Use a graphing utility to graph the region and verify your results. y
3x y
0, x
0, x
2

In Exercises 5760, find the area of the region bounded by the graphs. Use a graphing utility to graph the region and verify your results. y
1 x, y
0, x
0, x
4 y

In Exercises 5760, find the area of the region bounded by the graphs. Use a graphing utility to graph the region and verify your results. y
x 5
x, y
0, x
1, x
5 y

In Exercises 5760, find the area of the region bounded by the graphs. Use a graphing utility to graph the region and verify your results.y
3e y
0, x
2, x
1 x

In Exercises 6168, use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all x-values in the interval for which the function is equal to its average value. fx
6  x 2, 2 2 y

In Exercises 6168, use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all x-values in the interval for which the function is equal to its average value. fx
x  2x 0, 4 f

In Exercises 6168, use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all x-values in the interval for which the function is equal to its average value. fx
5e 0, 10 0.2x10 fx

In Exercises 6168, use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all x-values in the interval for which the function is equal to its average value. fx
2e 0, 4 x
4 f

In Exercises 6168, use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all x-values in the interval for which the function is equal to its average value.fx
x4  x 0, 2 2 f

In Exercises 6168, use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all x-values in the interval for which the function is equal to its average value.fx
0, 2 1 x  32 fx

In Exercises 6168, use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all x-values in the interval for which the function is equal to its average value. fx
0, 7 5x x2 1 fx

In Exercises 6168, use a graphing utility to graph the function over the interval. Find the average value of the function over the interval. Then find all x-values in the interval for which the function is equal to its average value. fx
0, 1 4x x2 1 fx

In Exercises 6972, state whether the function is even, odd, or neither. fx
3x 3  2x 4 f

In Exercises 6972, state whether the function is even, odd, or neither. gx
x fx
3x 3  2x 4 f

In Exercises 6972, state whether the function is even, odd, or neither.gt
2t 4 1 5  3t2 gx

In Exercises 6972, state whether the function is even, odd, or neither.ft
5t gt
2t 4 1 5

Use the value to evaluate each definite integral. Explain your reasoning. (a) (b) (c)

Use the value to evaluate each definite integral. Explain your reasoning. (a) (b) (c) 2 0 3x3 dx

Marginal Analysis In Exercises 7580, find the change in cost C, revenue R, or profit P, for the given marginal. In each case, assume that the number of units x increases by 3 from the specified value of x. dC dx
2.25 x
100

Marginal Analysis In Exercises 7580, find the change in cost C, revenue R, or profit P, for the given marginal. In each case, assume that the number of units x increases by 3 from the specified value of x. dC dx
20,000 x2 x
10

Marginal Analysis In Exercises 7580, find the change in cost C, revenue R, or profit P, for the given marginal. In each case, assume that the number of units x increases by 3 from the specified value of x. dR dx
48  3x x
12

Marginal Analysis In Exercises 7580, find the change in cost C, revenue R, or profit P, for the given marginal. In each case, assume that the number of units x increases by 3 from the specified value of x. dR dx
7520 900 x  x x
500

Marginal Analysis In Exercises 7580, find the change in cost C, revenue R, or profit P, for the given marginal. In each case, assume that the number of units x increases by 3 from the specified value of x.dP dx
400  x 150 x
200

Marginal Analysis In Exercises 7580, find the change in cost C, revenue R, or profit P, for the given marginal. In each case, assume that the number of units x increases by 3 from the specified value of x. dP dx
12.540  3x x x
125

Annuity In Exercises 8184, find the amount of an annuity with income function interest rate r, and term T. ct
$250, r
8%, T
6 years ct

Annuity In Exercises 8184, find the amount of an annuity with income function interest rate r, and term T. ct
$500, r
9%, T
4 years c

Annuity In Exercises 8184, find the amount of an annuity with income function interest rate r, and term T. ct
$1500, r
2%, T
10 years ct

Annuity In Exercises 8184, find the amount of an annuity with income function interest rate r, and term T. ct
$2000, r
3%, T
15 years c

Capital Accumulation In Exercises 8588, you are given the rate of investment Find the capital accumulation over a five-year period by evaluating the definite integral where t is the time in years. dI dt
500

Capital Accumulation In Exercises 8588, you are given the rate of investment Find the capital accumulation over a five-year period by evaluating the definite integral where t is the time in years. dI dt
100t

Capital Accumulation In Exercises 8588, you are given the rate of investment Find the capital accumulation over a five-year period by evaluating the definite integral where t is the time in years. dI dt
500t 1 d

Capital Accumulation In Exercises 8588, you are given the rate of investment Find the capital accumulation over a five-year period by evaluating the definite integral where t is the time in years. dI dt
12,000t t2 22 dI

Cost The total cost of purchasing and maintaining a piece of equipment for x years can be modeled by Find the total cost after (a) 1 year, (b) 5 years, and (c) 10 years.

Cost The total cost of purchasing and maintaining a piece of equipment for x years can be modeled by Find the total cost after (a) 1 year, (b) 5 years, and (c) 10 years.

Compound Interest A deposit of $2250 is made in a savings account at an annual interest rate of 12%, compounded continuously. Find the average balance in the account during the first 5 years.

Mortgage Debt The rate of change of mortgage debt outstanding for one- to four-family homes in the United States from 1993 through 2002 can be modeled by where M is the mortgage debt outstanding (in billions of dollars) and corresponds to 1993. In 1993, the mortgage debt outstanding in the United States was $3119 billion. (Source: Board of Governors of the Federal Reserve System) (a) Write a model for the debt as a function of t. (b) What was the average mortgage debt outstanding for 1993 through 2002?

Medicine The velocity v of blood at a distance r from the center of an artery of radius R can be modeled by where k is a constant. Find the average velocity along a radius of the artery. (Use 0 and R as the limits of integration.)

Biology The rate of change in the number of coyotes in a population is directly proportional to where t is time in years. When the population is 300, and when the population has increased to 500. (a) Find the population function. (b) Find the average number of coyotes over the first 5 years.

In Exercises 9598, use a symbolic integration utility to evaluate the definite integral. x 11  x dx 6 3 x 3x2  8 dx t

In Exercises 9598, use a symbolic integration utility to evaluate the definite integral. 1 1
2 x 11  x dx 6

In Exercises 9598, use a symbolic integration utility to evaluate the definite integral. 5 2  1 x2  1 x3 dx

In Exercises 9598, use a symbolic integration utility to evaluate the definite integral. 1 0 x3x3 13 dx 5