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Chapter 6: Q. 63 (page 512)

Use a definite integral to prove that a cone of radius r and height h has volume given by the formulaV=13蟺谤2h

Short Answer

Expert verified

It has been proved that volume of the cone is13蟺谤2h

Step by step solution

01

Step 1. Given information

The height of the cone is h.

The base radius is r

The volume to be proved isV=13蟺谤2h

02

Step 2. Proof of the volume

Lets draw a cone having given dimensions:

Here a circular slice of thickness dxhas been cut at distance of xfrom the base.

The radius of the slice is calculated as:

Since 鈭�ABC~鈭�AOE

Hence, BCr=h-xhBC=rh(h-x)=r-rhx

The volume of the slice is:

dv=蟺BC2dx=蟺r-rhx2dx=蟺r2+r2h2x2-2r2hxdx

The volume of the cone is:

V=鈭�0hdv=鈭�0h蟺r2+r2h2x2-2r2hxdx=蟺鈭�0hr2+r2h2x2-2r2hxdx=蟺r2x+r2h2.x33-2r22hx20h=蟺r2h+r2h33h2-r2h2h-0=蟺r2h+r2h3-r2h=13蟺谤2h

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dydx=xy+x+y+1,y(0)=c
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