Need help With MultiVariable Calculus Question Evaluate the iterated integral: " (2r?y + 3ry?) dr dy.

Name: Need help With MultiVariable Calculus Question Evaluate the iterated i
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Description: Need help With MultiVariable Calculus Question Evaluate the iterated integral: " (2r?y + 3ry?) dr dy.

The objective is to find the value of the iterated integral ∫-12∫242x2y+3xy2dxdy First integrate…

Answered: Evaluate the iterated integral: , S…

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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Need help With MultiVariable Calculus Question

Evaluate the iterated integral: " (2r?y + 3ry?) dr dy.

Expert Solution

Step 1

The objective is to find the value of the iterated integral $\int_{- 1}^{2} \int_{2}^{4} (2 x^{2} y + 3 x y^{2}) d x d y$

First integrate with respect to x,

$\begin{array}{rcl} \int_{- 1}^{2} \int_{2}^{4} (2 x^{2} y + 3 x y^{2}) d x d y & = & \int_{- 1}^{2} {(\frac{2 x^{3}}{3} y + \frac{3 x^{2}}{2} y^{2})}_{2}^{4} d y \\ = & \int_{- 1}^{2} (\frac{2 {(4)}^{3}}{3} y + \frac{3 {(4)}^{2}}{2} y^{2} - \frac{2 {(2)}^{3}}{3} y - \frac{3 {(2)}^{2}}{2} y^{2}) d y \\ = & \int_{- 1}^{2} (\frac{128}{3} y + \frac{48}{2} y^{2} - \frac{16}{3} y - \frac{12}{2} y^{2}) d y \\ = & \int_{- 1}^{2} (\frac{112}{3} y + 18 y^{2}) d y \end{array}$

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ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated