Answered: sketch the region R whose area is given…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

sketch the region R whose area is given by the iterated integral.

Then change the order of integration and show that both orders yield the same area.

Expert Solution

Step 1

The given integration $\int_{- 3}^{3} \int_{0}^{9 - y^{2}} d x d y$ .

We have to sketch the region and change the order of integration.

Step 2

The integration $\int_{- 3}^{3} \int_{0}^{9 - y^{2}} d x d y$ .

Here $x$ varies from $0$ to $9 - y^{2}$ and $y$ varies from $- 3$ to $3$ .

The region of the integration:

Step 3

The integration $\int_{- 3}^{3} \int_{0}^{9 - y^{2}} d x d y$ .

Solve the integration:

$\begin{array}{rcl} \int_{- 3}^{3} \int_{0}^{9 - y^{2}} d x d y & = & \int_{- 3}^{3} {[x]}_{0}^{9 - y^{2}} d y \\ = & \int_{- 3}^{3} [9 - y^{2}] d y \\ = & {[9 y - \frac{y^{3}}{3}]}_{- 3}^{3} \\ = & (9 (3) - \frac{3^{3}}{3}) - (9 (- 3) - \frac{{(- 3)}^{3}}{3}) \\ = & (27 - 9) - (- 27 + 9) \\ = & 18 - (- 18) \\ \int_{- 3}^{3} \int_{0}^{9 - y^{2}} d x d y & = & 36 \end{array}$

The value of the integral is $36$ .

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