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Chapter 13: Q. 31 (page 1083)

Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.

鈭02蟺鈭03鈭09-r2rdzdrd胃

Short Answer

Expert verified

鈭02蟺鈭03鈭09-r2rdzdrd胃=18蟺

Step by step solution

01

Given

We have to find the volume by using the triple integral.

鈭02蟺鈭03鈭09-r2rdzdrd胃
02

Evaluate the integral

V=鈭02蟺鈭03鈭09-r2rdzdrd胃V=鈭02蟺鈭03r9-r2drd胃

Substitute,

9-r2=t2-2rdr=2tdtdr=-trdtWhenr=0,t=9=3Whenr=3,t=9-9=0

Therefore,

V=-鈭02蟺鈭30t2dtd胃V=-鈭02蟺t3330d胃V=9鈭02蟺d胃V=9(2蟺-0)V=18蟺cubicunits

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