Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 15 - Multiple Integrals - 15.9 Change of Variables in Multiple Integrals - 15.9 Exercises - Page 1100: 8


The region is bounded by $y=1+x^2$ and the line $x=1$.

Work Step by Step

Let us consider $y=u(1+v^2) \implies u=\dfrac{y}{1+x^2}$ and $v=x$ Consider the given inequalities $0 \leq u \leq 1$ and $0 \leq v \leq 1$ This can be re-written as: $0 \leq \dfrac{y}{1+x^2} \leq 1$ and $0 \leq x \leq 1$ This gives: $0 \leq y \leq 1+x^2$ and $0 \leq x \leq 1$ Hence, the region is bounded by $y=1+x^2$ and the line $x=1$.
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