Area Between Two Curves

Math 5A

The Problem

Find the volume of the solid formed when the region bounded by y=sqrt(x) , x=4 and the x axis is revolved about the x axis.

€

x

The Idea Behind the Solution

Suppose we tried slicing this solid into 4 pieces by slicing perpendicular to the x axis.

Those four slices would look approximately like the four circular disks shown at the right, only the outer surface would not be as straight. We can approximate the volume of the solid by computing the volume of these four disks.

Finding the Volume of the Disks

The volume of each disk is r2h where h is the thickness of each disk (1 in this case, in general) and r is the functional value at a a point in the subinterval.

€

Δx

Volume for 4 disks

Approximation using 4 disks. For a better approximation, use more disks.

16 Disks.

64 Disks

1024 Disks – Approx. Volume 25.16

The Disk Method – Just add up the

€

€

V = limn→∞

π f (x)( )2Δx

i=1

n

∑ = π f (x)2( )dxa

b

∫€

r2Δx

For a solid formed by revolving a region bounded above by f(x) on [a,b] about the x axis…