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In Exercises 35–40, find the volume of the solid bounded above by the given function over the specified regionΩ. f(x,y)=4-x2-y2

Region:

Short Answer

Expert verified

Volume bounded by given function is4π.

Step by step solution

01

Step 1. Given Information

A function, f(x,y)=4-x2-y2

Region:

02

Step 2. Calculating the volume of the solid bounded above by the given function and region

The double integral is given by,

-2204-x24-x2-y2dydx+-20-4-x204-x2-y2dydx

Due to symmetry it can also be written as,

30204-x24-x2-y2dydx

Converting to polar coordinates, the integration becomes,

30π2024-r2rdrdθ

Put, 4-r2=t,-2rdr=dt,

We get, 30π240-t2dtdθ

=-320π223t3240dθ=-0π20-43240dθ=-30π2230-432dθ=80π21dθ=8θ0π2=8×π2=4π

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