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In Exercises 55-58, find the signed volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane.

localid="1651244649922" role="math" fx,y=x2-y3+4,

where localid="1651244677121" R=x,y|0x3and-2y3.

Short Answer

Expert verified

The required volume is 2254.

Step by step solution

01

Step 1. Given Information

We have given the following function :-

fx,y=x2-y3+4

where R=x,y|0x3and-2y3

We have to find the volume between the graph of the function fx,yand the specified rectangle in the xy-plane.

02

Step 2. To find Volume

We have given the following function :-

fx,y=x2-y3+4,

where localid="1650628129086" R=x,y|0x3and-2y3

Then the signed volume will be as following :-

Rx2-y3+4dA=-2303x2-y3+4dxdy

Then by using Fubini's theorem, we have :-

-2303x2-y3+4dxdy=-2303x2-y3+4dxdy

Then :-

localid="1651245884124" -2303x2-y3+4dxdy=-23x33-xy3+4x03dy=-23273-3y3+12-0dy=-239-3y3+12dy=-2321-3y3dy=21y-34y4-23=63-34×81--42-34×16=63-2434+42+12=117-2434=468-2434=2254

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