Answered: Using the transformation T: x = u + v,…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter7: Integration

Section7.5: The Area Between Two Curves

Problem 14E

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Question

Using the transformation T: x = u + v, y = u - v, the image of
the unit square S = {(u, v): 0 ≤ u ≤ 1, 0 ≤ v ≤ 1} is a region
R in the xy-plane. Explain how to change variables in the integral
∫∫_R ƒ(x, y) dA to find a new integral over S.

Expert Solution

Step 1

We have to write a new integral of the double integral $\iint_{R} f (x, y) d A$ over the unit square S using the

change of variables of x and y.

It is given that $x = u + v$ and $y = u - v$ . The change of variables of x and y transforms the region R into

the unit square S.

Step 2

The transformation used is $x = u + v$ and $y = u - v$ . We have to find $\frac{\partial (x, y)}{\partial (u, v)}$ .

Now,

$\begin{array}{rcl} \frac{\partial (x, y)}{\partial (u, v)} & = & |\begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} \end{matrix}| \\ = & |\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}| \\ = & (1) (- 1) - (1) (1) \\ = & - 1 - 1 \\ = & - 2 \\ ∴ \frac{\partial (x, y)}{\partial (u, v)} & = & - 2 \end{array}$

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