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Chapter 6: Q. 61 (page 513)

Use a definite integral to prove that the volume formula V=13蟺谤2hholds for a cone of radius 3 and height 5.

Short Answer

Expert verified

The volume of the cone is 15蟺cubic unit and it is proved by definite integral.

Step by step solution

01

Step 1. Given information

Height of the cone is 5and radius is 3

Formula of volume of the cone isV=13蟺谤2h

02

Step 2. Find volume of the cone by using given formula:

V=13蟺(3)2(5)=15蟺

03

Step 3. To prove the volume using integration.

Draw a 3d cone of the given dimension;

Here a circular slice has been cut at a distance of xfrom the base of the cone whose thickness is very small: dx

So the radius of the slice is calculated as:

Since, 鈭�ABC~鈭�ADE

Hence, BC3=5-x5BC=35(5-x)=3-35x

Area of the slice is:A=蟺(BC)2 =蟺3-35x2=蟺9+925x2-185x

Volume of the slice:

dv=蟺9+925x2-185xdx

So volume of the cone is:

V=鈭�05dv=鈭�05蟺9+925x2-185xdx=蟺9x+9x375-18x21050=蟺9脳5+9脳5375-18脳5210-0=蟺45+15-45=15蟺

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