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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Find the volume of the solid by subtracting two volumes.

38. The solid in the first octant under the plane z = x + y, above the surface z = xy, and enclosed by the surfaces x = 0, y = 0, and x2 + y2 = 4

To determine

To find: The volume of the solid that lies in between a plane and a surface.

Explanation

Given:

The plane and surface is respectively z=x+y,z=xy .

The Surfaces enclosed are, x=0,y=0,x2+y2=4 in the first octant.

Formula used:

The volume of the solid, V=Dz1dADz2dA , where, z1 and z2 are the given function.

Calculation:

Express the given plane equation as follows:

z1=x+y

And, z2=xy .

From the equation of the circle given, it is observed that x varies from −0 to 2 and y varies from 0 to 4x2 .

Thus, the volume of the solid is computed as follows.

V=Dz1dADz2dA=0204x2(x+y)dydx0204x2(xy)dydx=0204x2(x+yxy)dydx

First, compute the integral with respect to y.

V=02[xy+y22xy22]04x2dx

Apply the limit value for y,

V=02[(x4x2+(4x2)22x(4x2)22)(0+00)]dx=02x4x2dx+02(4x22)dx02x(4x2)2dx=02x4x2dx+02(4x22)dx02(4xx3)2dx

Compute the integral with respect to x

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