10S3X(xy+8) + dydxNow try the integral S¹⁰ S3 dady to see why theorder of integration is important.=X(xy+8)+

Answered: 10 S3 X (xy+8) + dydx Now try the…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Related questions

Question

10
S3
X
(xy+8) + dydx
Now try the integral S¹⁰ S3 dady to see why the
order of integration is important.
=
X
(xy+8)+

Expert Solution

Step 1

Consider:

$\begin{array}{rcl} \int_{3}^{7} \int_{4}^{10} \frac{x}{{(x y + 8)}^{4}} d y d x & = & \int_{3}^{7} x \int_{4}^{10} {(x y + 8)}^{- 4} d y d x \\ = & \int_{3}^{7} x \int_{4}^{10} \frac{{(x y + 8)}^{- 4 + 1}}{(- 4 + 1) x} d y d x \\ = & - \frac{1}{3} \int_{3}^{7} {[\frac{1}{{(x y + 8)}^{3}}]}_{4}^{10} d x \\ = & - \frac{1}{3} \int_{3}^{7} [\frac{1}{{(10 x + 8)}^{3}} - \frac{1}{{(4 x + 8)}^{3}}] d x \\ = & - \frac{1}{3} \int_{3}^{7} [{(10 x + 8)}^{- 3} - {(4 x + 8)}^{- 3}] d x \\ = & - \frac{1}{3} \int_{3}^{7} [\frac{{(10 x + 8)}^{- 3 + 1}}{- 2 \times 10} - \frac{{(4 x + 8)}^{- 3 + 1}}{- 2 \times 4}] d x \\ = & - \frac{1}{3} {[- \frac{1}{20 {(10 x + 8)}^{2}} + \frac{1}{8 {(4 x + 8)}^{2}}]}_{3}^{7} \\ = & - \frac{1}{3} ([- \frac{1}{20 {(10 (7) + 8)}^{2}} + \frac{1}{8 {(4 (7) + 8)}^{2}}] - [- \frac{1}{20 {(10 (3) + 8)}^{2}} + \frac{1}{8 {(4 (3) + 8)}^{2}}]) \\ = & - \frac{1}{3} ([- \frac{1}{20 {(78)}^{2}} + \frac{1}{8 {(36)}^{2}}] - [- \frac{1}{20 {(38)}^{2}} + \frac{1}{8 {(20)}^{2}}]) \\ = & 0.000063 \end{array}$