Volumes of solids with known cross sections

• You have already seen this being done in geometry.• For example, finding the volume of a prism , or a

cylinder:

• Find the area of the Base, and mutiply by the width or height.

The disk and washer method• We have also seen this in our study of Calculus

via Solids of revolution• The disk and washer method use Circular

cross sections• We multiply the cross sections times the width

to get the actual volume.

It can really be ANY cross section

• It doesn’t necessarily have to be circular.• It can be a square, triangle, ellipse, etc.. • If we can find the Area of the cross section,

then we can multiple it times the width (just like before) to get the entire volume.

• ANIMATION

Formulas

• When the cross section is perpendicular to the x-axis, integrate with respect to x:

• When the cross section is perpendicular to the y-axis, integrate with respect to y:

b

a

dxxAV )(

b

a

dyyAV )(

Example:

• A solid is formed with a base bounded by the graphs of:

Find the volume of the solid using equilateral triangle cross – sections taken perpendicular to the x-axis.

21)(x

xf 0x 21)(x

xg

Practice