2.A. Rewrite the order of integration.\[\int_{0}^{\sqrt{3}/2} \int_{\arcsin x}^{\pi/3} f(x,y) \, dy \, dx\]B. Find the volume of the solid under the surface $ z = 9 - x^2 + 4y^2 $ and above the rectangle $ R = [1, 3] \times [0, 4] $.

Answered: Aa Rewrite the order of integration.…

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

2.

A. Rewrite the order of integration.

\[
\int_{0}^{\sqrt{3}/2} \int_{\arcsin x}^{\pi/3} f(x,y) \, dy \, dx
\]

B. Find the volume of the solid under the surface $ z = 9 - x^2 + 4y^2 $ and above the rectangle $ R = [1, 3] \times [0, 4] $.

Expert Solution

Step 1

A. Here, we have to rewrite the order of integration, for the given integral.
$\int_{0}^{\sqrt{3} / 2} \int_{a r c \sin x}^{π / 3} f (x, y) d y d x$
From the given integral we get the limits for variable y and x as:

Limits of y : $y = a r c \sin x \to y = \frac{π}{3} or \sin y = x \to y = \frac{π}{3}$ ;
Limits of x : $x = 0 \to x = \frac{\sqrt{3}}{2}$ ;

We can plot these limits on a X-Y plane and get the required region as

Step 2

The given order of integration is dy dx. We have too re-write the limits in the order dx dy.

So, first we evaluate the limits of x and then y.

Limits of x : $x = 0 \to x = \sin y$
Limits of y : $y = 0 \to y = \frac{π}{3}$

So, our integral changes to $\int_{0}^{\frac{π}{3}} \int_{0}^{\sin y} f (x, y) d x d y$

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