Give an example of a region that cannot be expressed by either of the forms shown in Figure 34. (One example is the disk with a hole in the middle between the graphs of $x^{2}+y^{2}=1$ and $x^{2}+y^{2}=2$ in Figure $10 . )$

The idea of the average value of a function, discussed earlier for functions of the form $y=f(x)$ , can be extended to functions of more than one independent variable. For a function $z=f(x, y)$ ,

the average value of $f$ over a region $R$ is defined as

$\frac{1}{A} \iint_{R} f(x, y) d x d y$

where $A$ is the area of the region $R .$ Find the average value foreach function over the regions $R$ having the given boundaries.

## Discussion

## Video Transcript

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## Recommended Questions

Evaluate each double integral. If the function seems too difficult to integrate, try interchanging the limits of integration, as in Exercises 37 and $38 .$

$$

\int_{0}^{\ln 2} \int_{e^{y}}^{2} \frac{1}{\ln x} d x d y

$$

Evaluate each double integral. If the function seems too difficult to integrate, try interchanging the limits of integration, as in Exercises 37 and $38 .$

$$\int_{0}^{2} \int_{y / 2}^{1} e^{x^{2}} d x d y$$

Although it is often true that a double integral can be evaluated by using either $d x$ or $d y$ first, sometimes one choice over the other makes the work easier. Evaluate the double integrals in Exercises 37 and 38 in the easiest way possible.

$$

\iint_{R} 2 x^{3} e^{x^{2} y} d x d y ; \quad 0 \leq x \leq 1,0 \leq y \leq 1

$$

Converting the Limits of Integration In Exercises 37-42, evaluate the definite integral using (a) the given integration limits and (b) the limits obtained by trigonometric substitution.

$$\int_{0}^{3} \frac{x^{3}}{\sqrt{x^{2}+9}} d x$$

Although it is often true that a double integral can be evaluated by using either $d x$ or $d y$ first, sometimes one choice over the other makes the work easier. Evaluate the double integrals in Exercises 37 and 38 in the easiest way possible.

$$\iint_{R} x e^{x y} d x d y ; \quad 0 \leq x \leq 2,0 \leq y \leq 1$$

Finding Integrals Evaluate the integrals in Exercises 29-50.

$$\int\left(2 e^{x}-3 e^{-2 x}\right) d x$$

Converting the Limits of Integration In Exercises 37-42, evaluate the definite integral using (a) the given integration limits and (b) the limits obtained by trigonometric substitution.

$$\int_{4}^{8} \frac{\sqrt{x^{2}-16}}{x^{2}} d x$$

Evaluate each double integral.

$$

\int_{0}^{2} \int_{0}^{3 y}\left(x^{2}+y\right) d x d y

$$

In Exercises $21-32,$ express the integrand as a sum of partial fractions

and evaluate the integrals.

$$

\int \frac{y^{2}+2 y+1}{\left(y^{2}+1\right)^{2}} d y

$$

Use the results of Equations $(2)$ and $(4)$ to evaluate the integrals in

Exercises $29-40 .$

$$

\int_{0.5}^{2.5} x d x

$$

Use the results of Equations $(2)$ and $(4)$ to evaluate the integrals in

Exercises $29-40 .$

$$

\int_{a}^{2 a} x d x

$$

Use the results of Equations $(2)$ and $(4)$ to evaluate the integrals in

Exercises $29-40 .$

$$

\int_{1}^{\sqrt{2}} x d x

$$

Use any method to evaluate the integrals in Exercises $15-38 .$ Most will require trigonometric substitutions, but some can be evaluated by other methods.

$$\int_{0}^{\sqrt{3} / 2} \frac{4 x^{2} d x}{\left(1-x^{2}\right)^{3 / 2}}$$

In Exercises $33-46,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.

$$\int_{0}^{3 / 2} \int_{0}^{9-4 x^{2}} 16 x d y d x$$

Use the results of Equations $(2)$ and $(4)$ to evaluate the integrals in

Exercises $29-40 .$

$$

\int_{a}^{\sqrt{3}} x d x

$$

Evaluating integrals Evaluate the following integrals as they are written.

$$\int_{0}^{\ln 2} \int_{e^{x}}^{2} d y d x$$

Evaluating integrals Evaluate the following integrals as they are written.

$$\int_{0}^{1} \int_{0}^{x} 2 e^{x^{2}} d y d x$$

In Exercises $33-38,$ perform long division on the integrand, write the

proper fraction as a sum of partial fractions, and then evaluate the

integral.

$$

\int \frac{y^{4}+y^{2}-1}{y^{3}+y} d y

$$

In the following exercises, calculate the integrals by interchanging the order of integration.

$$\int_{1}^{9}\left(\int_{4}^{2} \frac{\sqrt{x}}{y^{2}} d y\right) d x$$

Evaluating integrals Evaluate the following integrals.

$$\int_{0}^{1} \int_{0}^{x} 2 e^{x} d y d x$$

Use the results of Equations $(2)$ and $(4)$ to evaluate the integrals in

Exercises $29-40 .$

$$

\int_{0}^{3 b} x^{2} d x

$$

Use any method to evaluate the integrals in Exercises $15-38 .$ Most will require trigonometric substitutions, but some can be evaluated by other methods.

$$\int \frac{x}{\sqrt{9-x^{2}}} d x$$

Use the results of Equations $(2)$ and $(4)$ to evaluate the integrals in

Exercises $29-40 .$

$$

\int_{0}^{\sqrt[3]{7}} x^{2} d x

$$

Evaluating integrals Evaluate the following integrals.

$$\int_{0}^{\ln 2} \int_{e}^{2} d y d x$$

Converting the Limits of Integration In Exercises 37-42, evaluate the definite integral using (a) the given integration limits and (b) the limits obtained by trigonometric substitution.

$$\int_{0}^{\sqrt{3 / 2}} \frac{1}{\left(1-t^{2}\right)^{5 / 2}} d t$$

Evaluate each double integral.

$$\int_{2}^{4} \int_{2}^{x^{2}}\left(x^{2}+y^{2}\right) d y d x$$

Evaluating integrals Evaluate the following integrals.

$$\int_{0}^{\ln 2} \int_{-x}^{2} \frac{y}{x} d x d y$$

Evaluating integrals Evaluate the following integrals as they are written.

$$\int_{-2}^{2} \int_{x^{2}}^{8-x^{2}} x d y d x$$

Evaluating integrals Evaluate the following integrals.

$$\int_{0}^{4} \int_{-\sqrt{16-y}}^{\sqrt{16-y^{2}}} 2 x y d x d y$$

Evaluating integrals Evaluate the following integrals.

$$\int_{0}^{1} \int_{0}^{x} 2 e^{x^{2}} d y d x$$

Converting the Limits of Integration In Exercises 37-42, evaluate the definite integral using (a) the given integration limits and (b) the limits obtained by trigonometric substitution.

$$\int_{0}^{3 / 5} \sqrt{9-25 x^{2}} d x$$

In Exercises $37-40,$ (a) find the points of discontinuity of the integrand on the interval of integration, and (b) use area to evaluate the integral.

$$\int_{-2}^{3} \frac{x}{|x|} d x$$

In Exercises $33-38,$ perform long division on the integrand, write the

proper fraction as a sum of partial fractions, and then evaluate the

integral.

$$

\int \frac{2 y^{4}}{y^{3}-y^{2}+y-1} d y

$$

In the following exercises, calculate the integrals by interchanging the order of integration.

$$\int_{\ln 2}^{\ln 3}\left(\int_{0}^{1} e^{x+y} d y\right) d x$$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2 .

$$

\int_{-2}^{3}\left(x^{2}+3 x-5\right) d x

$$

In the following exercises, evaluate each definite integral using the Fundamental Theorem of Calculus, Part 2.

$$

\int_{-2}^{3}\left(x^{2}+3 x-5\right) d x

$$

Evaluate the following integrals using integration by parts.

$$\int_{0}^{\ln 2} x e^{x} d x$$

Converting the Limits of Integration In Exercises 37-42, evaluate the definite integral using (a) the given integration limits and (b) the limits obtained by trigonometric substitution.

$$\int_{0}^{\sqrt{3} / 2} \frac{t^{2}}{\left(1-t^{2}\right)^{3 / 2}} d t$$

In Exercises $33-46,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.

$$\int_{0}^{3} \int_{1}^{e^{y}}(x+y) d x d y$$

Evaluating integrals Evaluate the following integrals.

$$\int_{0}^{\pi / 2} \int_{0}^{\cos y} e^{\sin y} d x d y$$

Consider the integral $\int_{1}^{3} \int_{-1}^{1}\left(2 y^{2}+x y\right) d y d x$. Give the variable of integration in the first (inner) integral and the limits of integration. Give the variable of integration in the second (outer) integral and the limits of integration.

Finding Integrals Evaluate the integrals in Exercises 29-50.

$$\int_{\ln 4}^{\ln 9} e^{x / 2} d x$$

Use the results of Equations $(2)$ and $(4)$ to evaluate the integrals in

Exercises $29-40 .$

$$

\int_{0}^{\sqrt[3]{b}} x^{2} d x

$$

Evaluate the following integrals two ways.

a. Simplify the integrand first and then integrate.

b. Change variables (let $u=\ln x$ ), integrate, and then simplify your answer. Verify that both methods give the same answer.

$$\int_{1}^{\sqrt{3}} \frac{\operatorname{sech}(\ln x)}{x} d x$$

In Exercises $21-32,$ express the integrand as a sum of partial fractions

and evaluate the integrals.

$$

\int \frac{8 x^{2}+8 x+2}{\left(4 x^{2}+1\right)^{2}} d x

$$

In Exercises $33-46,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.

$$\int_{0}^{\ln 2} \int_{e^{y}}^{2} d x d y$$

Use a CAS double-integral evaluator to find the integrals in Then reverse the order of integration and evaluate, again with a CAS.

$$\int_{1}^{2} \int_{y^{3}}^{8} \frac{1}{\sqrt{x^{2}+y^{2}}} d x d y$$

In Exercises $21-32,$ express the integrand as a sum of partial fractions

and evaluate the integrals.

$$

\int \frac{1}{x^{4}+x} d x

$$

Use a change of variables to evaluate the following integrals.

$$\int_{\sqrt{2}}^{\sqrt{3}}(x-1)\left(x^{2}-2 x\right)^{11} d x$$

In Exercises $33-46,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.

$$\int_{1}^{e} \int_{0}^{\ln x} x y d y d x$$

Use the result of Exercise 40 part (c) to evaluate the following integrals.

(a) $ \displaystyle \int_0^\infty x^2 e^{-x^2}\ dx $ (b) $ \displaystyle \int_0^\infty \sqrt{x}\ e^{-x}\ dx $

In the following exercises, calculate the integrals by interchanging the order of integration.

$$\int_{1}^{16}\left(\int_{1}^{8} \int_{1}^{8}\left(\sqrt[4]{x}+2^{3} y\right) d y\right) d x$$

Use the value of the first integral I to evaluate the two given integrals.Using properties of integrals Use the value of the first integral I to evaluate the two given integrals.

$$I=\int_{0}^{1}\left(x^{3}-2 x\right) d x=-\frac{3}{4}$$

a. $\int_{0}^{1}\left(4 x-2 x^{3}\right) d x$

b. $\int_{1}^{0}\left(2 x-x^{3}\right) d x$

Use any method to evaluate the integrals in Exercises $15-38 .$ Most will require trigonometric substitutions, but some can be evaluated by other methods.

$$\int \frac{x d x}{25+4 x^{2}}$$

In Exercises $33-38,$ perform long division on the integrand, write the

proper fraction as a sum of partial fractions, and then evaluate the

integral.

$$

\int \frac{x^{4}}{x^{2}-1} d x

$$

In Exercises $33-46,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.

$$\int_{0}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} 6 x d y d x$$

Evaluate each double integral.

$$\int_{0}^{4} \int_{1}^{e^{x}} \frac{x}{y} d y d x$$

In Exercises $27-40$ , evaluate each integral using Part 2 of the Fundamental Theorem. Support your answer with NINT if you are unsure.

$$\int_{-2}^{-1} \frac{2}{x^{2}} d x$$

Evaluating integrals Evaluate the following integrals as they are written.

$$\int_{0}^{2} \int_{x^{2}}^{2 x} x y d y d x$$

Evaluate the integrals in Exercises $41-44$ by changing the order of

integration in an appropriate way.

$$

\int_{0}^{2} \int_{0}^{4-x^{2}} \int_{0}^{x} \frac{\sin 2 z}{4-z} d y d z d x

$$

In Exercises $33-38,$ perform long division on the integrand, write the

proper fraction as a sum of partial fractions, and then evaluate the

integral.

$$

\int \frac{2 x^{3}-2 x^{2}+1}{x^{2}-x} d x

$$

Evaluating a Definite Integral In Exercises 21-24, use partial fractions to evaluate the definite

integral. Use a graphing utility to verify your result.

$$\int_{1}^{5} \frac{x-1}{x^{2}(x+1)} d x$$

Finding Integrals Evaluate the integrals in Exercises 29-50.

$$\int 8 e^{(x+1)} d x$$

Use a CAS double-integral evaluator to find the integrals in Exercises 89.94 . Then reverse the order of integration and evaluate, again with a CAS.

$$\int_{0}^{2} \int_{0}^{4-y^{2}} e^{x y} d x d y$$

Use any method to evaluate the integrals in Exercises $15-38 .$ Most will require trigonometric substitutions, but some can be evaluated by other methods.

$$\int \frac{\left(1-x^{2}\right)^{3 / 2}}{x^{6}} d x$$

Evaluate the integrals. Some integrals do not require integration by parts.

$$

\int \frac{1}{x(\ln x)^{2}} d x

$$

Evaluate the integrals. Some integrals do not require integration by parts.

$$\int \frac{1}{x(\ln x)^{2}} d x$$

Evaluating a Definite Integral In Exercises $43-52,$ evaluate the definite integral. Use a graphing utility to verify your result.

$$\int_{0}^{3} x e^{x / 2} d x$$

In Exercises $33-46,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.

$$\int_{0}^{1} \int_{1}^{e^{x}} d y d x$$

In the following exercises, calculate the integrals by interchanging the order of integration.

$$\int_{1}^{6}\left(\int_{2}^{9} \frac{\sqrt{y}}{x^{2}} d y\right) d x$$

Express the integrand as a sum of partial fractions and evaluate the integrals.

$$\int \frac{x^{2} d x}{(x-1)\left(x^{2}+2 x+1\right)}$$

Two ways Evaluate the following integrals two ways.

a. Simplify the integrand first, and then integrate.

b. Change variables (let $u=\ln x$ ), integrate, and then simplify your answer. Verify that both methods give the same answer.

$$\int_{1}^{\sqrt{3}} \frac{\operatorname{sech}(\ln x)}{x} d x$$

Use any method to evaluate the integrals in Exercises $55-66$

$$

\int \frac{2^{x}-2^{-x}}{2^{x}+2^{-x}} d x

$$

In Exercises $37-40,$ (a) find the points of discontinuity of the integrand on the interval of integration, and (b) use area to evaluate the integral.

$$\int_{-6}^{5} 2 \operatorname{int}(x-3) d x$$

Definite integrals Use a computer algebra system to evaluate the following definite integrals. In each case, find an exact value of the integral (obtained by a symbolic method) and find an approximate value (obtained by a numerical method). Compare the results.

$$\int_{0}^{4}\left(9+x^{2}\right)^{3 / 2} d x$$

In the following exercises, calculate the integrals by interchanging the order of integration.

$$\int_{-1}^{1}\left(\int_{-2}^{2}(2 x+3 y+5) d x\right) d y$$

Finding Integrals Evaluate the integrals in Exercises 29-50.

$$\int_{\ln 2}^{\ln 3} e^{x} d x$$