Answered: e dy dx

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter5: Exponential And Logarithmic Functions

Section5.5: Properties Of Logarithms

Problem 118E

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Question

sketch the region of integration, reverse the order
of integration, and evaluate the integral.

Expert Solution

Step 1

Given:

$\int_{0}^{3} \int_{\sqrt{x / 3}}^{1} e^{y^{3}} d y d x$

To Find:

$\begin{array}{l} (a) . Sketch the region of integration. \\ (b) . Reverse the order of integration. \\ (c) . Evaluate the integral. \end{array}$

Step 2

Explanation:

(a).

$\begin{array}{l} \int_{0}^{3} \int_{\sqrt{\frac{x}{3}}}^{1} e^{y^{3}} d y d x \\ From above we have: \\ \sqrt{\frac{x}{3}} \leq y \leq 1 and 0 \leq x \leq 3. \end{array}$

$Hence, the corresponding region is$ :

Step 3

(b).

$\begin{array}{l} y = \sqrt{\frac{x}{3}} \\ \Rightarrow y^{2} = \frac{x}{3} \\ \Rightarrow x = 3 y^{2} \end{array}$

$\begin{array}{l} from above we get: \\ 0 \leq y \leq 1 \\ and 0 \leq x \leq 3 y^{2} \\ \Rightarrow \int_{0}^{3} \int_{\sqrt{\frac{x}{3}}}^{1} e^{y^{3}} d y d x = \int_{0}^{1} \int_{0}^{3 y^{2}} e^{y^{3}} d x d y \end{array}$

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