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Problem 37

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.

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

Answer

$=e^{2}-3$

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## Discussion

## Video Transcript

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## 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

$$

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.

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

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.

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

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

$$

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.

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

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.

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

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.

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

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.

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

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