As the equation is exact, there is a function f such that. = M(x, y) and = N(x, y). af ax af ду To find the function f, first take the integral of M with respect to x. Note the middle term requires the use of integration by parts. af əx -J = [M(x, y) dx = f(x- (y - 8xe* - 6x²) dx +(([ = xy + f(x, y) = dx -(C xe ) - 2x3 -2x³ + h(y)

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
ChapterA: Appendix
SectionA.2: Geometric Constructions
Problem 10P: A soda can has a volume of 25 cubic inches. Let x denote its radius and h its height, both in...
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As the equation is exact, there is a function f such that = M(x, y) and = N(x, y).
af
əx
af
ay
To find the function f, first take the integral of M with respect to x. Note the middle term requires the use of integration by parts.
= [ of dx
f(x, y) =
=
M(x, y) dx
-for
- 8xe* - 6x²) dx
= xy +
+(([
1xex) - 2
The unknown function h in the variable y functions as a constant of integration in the integration with respect to x. When taking the partial derivative of f(x, y) with respect to x, the derivative of the term h(y) is 0.
xex-2x³ + h(y)
Transcribed Image Text:As the equation is exact, there is a function f such that = M(x, y) and = N(x, y). af əx af ay To find the function f, first take the integral of M with respect to x. Note the middle term requires the use of integration by parts. = [ of dx f(x, y) = = M(x, y) dx -for - 8xe* - 6x²) dx = xy + +(([ 1xex) - 2 The unknown function h in the variable y functions as a constant of integration in the integration with respect to x. When taking the partial derivative of f(x, y) with respect to x, the derivative of the term h(y) is 0. xex-2x³ + h(y)
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