Thomas' Calculus 13th Edition

Published by Pearson

Chapter 15: Multiple Integrals - Section 15.2 - Double Integrals over General Regions - Exercises 15.2 - Page 882: 14

Answer

a) $\int_0^{\pi/4} \int_{\tan x}^{1} f(x,y) dy dx$ b)$\int_{0}^{1} \int_{0}^{\tan^{-1} y} f(x,y) dx dy$

Work Step by Step

(a) For vertical cross-sections, the region $R$ can be defined as: $R=$ { $( x,y) | \tan x \leq y \leq 1 , 0 \leq x \leq \dfrac{\pi}{4}$} Hence, we have $\int_0^{\pi/4} \int_{\tan x}^{1} f(x,y) dy dx$ (b) For horizontal cross-sections, the region $R$ can be defined as: $R=$ { $( x,y) | 0 \leq x \leq \tan^{-1} y , 0 \leq y \leq 1$} Hence, we have $\int_{0}^{1} \int_{0}^{\tan^{-1} y} f(x,y) dx dy$

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