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

## Discussion

## Video Transcript

No transcript available

## Recommended Questions

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

$$

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.

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

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 $38 .$

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

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral.

$$\iint_{R} \frac{x}{(1+x y)^{2}} d A ; R=\{(x, y): 0 \leq x \leq 4,1 \leq y \leq 2\}$$

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral.

$$\iint_{R}(y+1) e^{x(y+1)} d A ; R=\{(x, y): 0 \leq x \leq 1,-1 \leq y \leq 1\}$$

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral.

$$\iint_{R} y^{3} \sin x y^{2} d A ; R=\{(x, y): 0 \leq x \leq 2,0 \leq y \leq \sqrt{\pi / 2}\}$$

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral.

$$\iint_{R} x \sec ^{2} x y d A ; R=\{(x, y): 0 \leq x \leq \pi / 3,0 \leq y \leq 1\}$$

Improper double integrals can often be computed similarly to im- proper integrals of one variable. The first iteration of the following improper integrats is conducted just as if they were proper integrals.

One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section $8.7 .$ Evaluate the improper integrals in Exercises $69-72$ as iterated integrals.

$$\int_{0}^{\infty} \int_{0}^{\infty} x e^{-(x+2 y)} d x d y$$

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section 8.7. Evaluate the improper integrals as iterated integrals.

$$\int_{1}^{\infty} \int_{e^{-x}}^{1} \frac{1}{x^{3} y} d y d x$$

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section $8.8 .$ Evaluate the improper integrals as iterated integrals.

\begin{equation}\int_{0}^{\infty} \int_{0}^{\infty} x e^{-(x+2 y)} d x d y\end{equation}

Improper double integrals can often be computed similarly to im- proper integrals of one variable. The first iteration of the following improper integrats is conducted just as if they were proper integrals.

One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section $8.7 .$ Evaluate the improper integrals in Exercises $69-72$ as iterated integrals.

$$\int_{1}^{\infty} \int_{e^{-y}}^{1} \frac{1}{x^{3} y} d y d x$$

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section 8.7. Evaluate the improper integrals as iterated integrals.

$$\int_{-1}^{1} \int_{-1 / \sqrt{1-x^{2}}}^{1 / \sqrt{1-x^{2}}}(2 y+1) d y d x$$

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral.

$$\iint_{R} y \cos x y d A ; R=\{(x, y): 0 \leq x \leq 1,0 \leq y \leq \pi / 3\}$$

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral.

$$\iint_{K} 6 x^{5} e^{x^{3} y} d A ; R=\{(x, y): 0 \leq x \leq 2,0 \leq y \leq 2\}$$

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section $8.8 .$ Evaluate the improper integrals as iterated integrals.

\begin{equation}\int_{-1}^{1} \int_{-1 / \sqrt{1-x^{2}}}^{1 / \sqrt{1-x^{2}}}(2 y+1) d y d x\end{equation}

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section $8.8 .$ Evaluate the improper integrals in Exercises $69-72$ as iterated integrals.

\begin{equation}\int_{1}^{\infty} \int_{e^{-x}}^{1} \frac{1}{x^{3} y} d y d x\end{equation}

In the following exercises, evaluate the double integral $\iint_{D} f(x, y) d A$ over the region $D .$

$f(x, y)=x y$ and $D=\left\{(x, y) |-1 \leq y \leq 1, y^{2}-1 \leq x \leq \sqrt{1-y^{2}}\right\}$

Improper double integrals can often be computed similarly to im- proper integrals of one variable. The first iteration of the following improper integrats is conducted just as if they were proper integrals.

One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section $8.7 .$ Evaluate the improper integrals in Exercises $69-72$ as iterated integrals.

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

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.

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

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

\begin{equation}

\int_{0}^{1} \int_{2}^{4-2 x} d y d x

\end{equation}

Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why it's easier.

$ \displaystyle \iint\limits_D y^2 e^{xy}\ dA $, $ D $ is bounded by $ y = x $, $ y = 4 $, $ x = 0 $

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

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

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 geometry or symmetry, or both, to evaluate the double integral.

$ \displaystyle \iint\limits_D (x + 2)\ dA $

$ D = \{(x, y) \mid 0 \le y \le \sqrt{9 - x^2} \} $

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, evaluate the double integral $\iint_{D} f(x, y) d A$ over the region $D .$

$f(x, y)=2 x+5 y$ and $D=\left\{(x, y) | 0 \leq x \leq 1, x^{3} \leq y \leq x^{3}+1\right\}$

Use a CAS double-integral evaluator to estimate the values of the integrals.

$$\int_{1}^{3} \int_{1}^{x} \frac{1}{x y} d y d x$$

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

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

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

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

Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why it's easier.

$ \displaystyle \iint\limits_D y\ dA $, $ D $ is bounded by $ y = x - 2 $, $ x = y^2 $

Use geometry or symmetry, or both, to evaluate the double integral.

$ \displaystyle \iint\limits_D (2 + x^2 y^3 - y^2 \sin x)\ dA $

$ D = \{(x, y) \mid | x | + | y | \le 1 \} $

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

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

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.

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

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

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

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

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

In Exercises $13-20$ , evaluate the double integral over the given region $R .$

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

In the following exercises, evaluate the double integral $\iint_{D} f(x, y) d A$ over the region $D .$

$f(x, y)=1$ and $D=\left\{(x, y) | 0 \leq x \leq \frac{\pi}{2}, \sin x \leq y \leq 1+\sin x\right\}$

Use a CAS double-integral evaluator to estimate the values of the integrals.

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

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

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

Evaluate the double integral.

$ \displaystyle \iint\limits_D y \sqrt{x^2 - y^2 }\ dA $, $ D = \{(x, y) \mid 0 \le x \le 2, 0 \le y \le x \} $

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section $8.8 .$ Evaluate the improper integrals as iterated integrals.

\begin{equation}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{\left(x^{2}+1\right)\left(y^{2}+1\right)} d x d y\end{equation}

Evaluate each double integral.

$$

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

$$

Apply Green's Theorem to evaluate the integrals in Exercises $27-30 .$

$$\oint(3 y d x+2 x d y)$$

$C:$ The boundary of $0 \leq x \leq \pi, 0 \leq y \leq \sin x$

Use a CAS double-ntegral evaluator to estimate the values of the integrals in Exercises $85-88 .$

$$\int_{-1}^{1} \int_{0}^{\sqrt{1-x^{2}}} 3 \sqrt{1-x^{2}-y^{2}} d y d x$$

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

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

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

$$\int_{0}^{\pi / 6} \int_{\sin x}^{1 / 2} x y^{2} d y d x$$

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

$$\int_{0}^{\sqrt{3}} \int_{0}^{\tan ^{-1} y} \sqrt{x y} d x d y$$

In evaluating a double integral over a region $ D $, a sum of iterated integrals was obtained as follows:

$$ \displaystyle \iint\limits_D f(x, y)\ dA = \int_0^1 \int_0^{2y} f(x, y)\ dx dy + \int_1^3 \int_0^{3 - y} f(x, y)\ dx dy $$

Sketch the region $ D $ and express the double integral as an iterated integral with reversed order of integration.

Evaluate the integrals.

$$\int_{0}^{\sqrt{2}} \int_{0}^{3 y} \int_{x^{2}+3 y^{2}}^{8-x^{2}-y^{2}} d z d x d y$$

The integrals in Exercises $1-40$ are in no particular order. Evaluate

each integral using any algebraic method or trigonometric identity

you think is appropriate, and then use a substitution to reduce it to a

standard form.

$$

\int \frac{\ln y}{y+4 y \ln ^{2} y} 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.

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

Evaluating integrals Evaluate the following integrals as they are written.

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

Evaluate each double integral.

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

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

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

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

$$

Use a CAS double-integral evaluator to estimate the values of the integrals in Exercises $85-88 .$

\begin{equation}\int_{1}^{3} \int_{1}^{x} \frac{1}{x y} d y d x\end{equation}

Calculate the double integral.

$\iint_{R} y e^{-x y} d A, \quad R=[0,2] \times[0,3]$

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

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