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Chapter 6: Q. 62 (page 513)

Use a definite integral to prove that the volume formula V=43蟺谤3holds for a sphere of radius 3.

Short Answer

Expert verified

The volume of the sphere is36蟺by suing formula and integration also.

Step by step solution

01

Step 1. Given information

The formula for the volume of sphere is V=43蟺谤3, where ris tthe radius

The radius is 3.

02

Step 2. Find volume of sphere using formula:

V=43蟺33=36蟺

03

Step 3. Volume  of sphere using integration

Draw a sphere of radius 3:

A circular slice of thickness dxhas been cut at a distance of xfrom the center of the sphere.

The radius of the circular slice is:

AB=32-x2

So the area of the slice is:A=蟺AB2=蟺(32-x2)=蟺(9-x2)

The volume of the slice is dV=Adx=蟺9-x2dx

Hence the volume of the sphere is:

V=鈭�-33dV=鈭�-33蟺9-x2dx=蟺鈭�-339-x2dx=2蟺鈭�039-x2dx=2蟺9x-x3303=2蟺9脳3-333-0=2蟺27-9=36蟺

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