Multivariable Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 0-53849-787-4
ISBN 13: 978-0-53849-787-9

Chapter 15 - Multiple Integrals - 15.10 Exercises - Page 1071: 9

Answer

The region is bounded by the line $y=1$, the y-axis and $y=\sqrt x$.

Work Step by Step

Need to consider the following few cases in order to get the region. 1) First case : From $(0,0)$ to $(0,1)$. Then, $u=0; v=t$ where $0 \leq t \leq 1$ Under the transformation $x=u^2, y=v$, thus, the parametric equations are: $x=0,y=t$ where $0 \leq t \leq 1$ 2) Second case : From $(0,1)$ to $(1,1)$. Then, we have $u=t; v=1$ where $0 \leq t \leq 1$ Under the transformation $x=u^2, y=v$, thus, the parametric equations are: $x=t^2,y=1$ where $0 \leq t \leq 1$ 3) Third case: From $(1,1)$ to $(0,0)$. Then, we have $u=1-t; v=1-t$ where $0 \leq t \leq 1$ Under the transformation $x=u^2, y=v$, thus, the parametric equations are: $x=(1-t)^2,y=1-t$ where $0 \leq t \leq 1$ This gives: $x=y^2 \implies y=\sqrt x$ Hence, the region is bounded by the line $y=1$, the y-axis and $y=\sqrt x$.
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