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Q. 22

Found in: Page 1119


Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

Find the area of S is the portion of the plane with equation x = y + z that lies above the region in the xy-plane that is bounded by y = x, y = 5, y = 10, and the y-axis.

Hence the area of triangle is =7532

See the step by step solution

Step by Step Solution

Step 1. Given

Equation of plane x=y+ z

Step 2.  Formula for finding the area of surface

If a surface S is given by z=f(x,y) for f(x,y)DR2Then the surface area of smooth surface is SdS=Szx2+zy2+1dA=Dzx2+zy2+1dA(1

Step 3.  Finding the partial derivative 

x=y+zthen,z=x-ynow, first find zx=xx-y=1andzy=yx-y=-1

Step 4. Graph

viewed it as a x-simple, then the region of integration will be

D=x,y|0xy, 5y10

Step 5. Finding the area

The area of surface is SdS=Dzx2+zy2+1dA=5100y12+-12+1dxdy=5100y3dxdy=35100ydxdy=35100ydxdy=3510[x]0ydy=3510[y0]dy=3510ydy=3510ydy=2[y22]510=2[1022-522]=7532Hence the area of triangle is =7532

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