Volumes of Solids with
Known Cross Sections
Ms. Battaglia
AP Calculus
Write an equation to give the
area of each shape.
Square
Rectangle that is three times
as tall as it is wide
Equilateral Triangle
Isosceles Triangle that is twice
as high as its base
Write an equation to give the
area of each shape.
Semicircle
Rectangle that is twice
as wide as it is tall
Isosceles Triangle
Rectangle that is two-thirds
as tall as it is wide
Write an equation to give the
area of each shape in terms of x.
Quarter Circle
Quarter Circle
Isosceles triangle that is
twice as tall as it is wide
Equilateral Triangle
Steps for Finding Volume of a
Solid with Known Cross Sections
• Draw the graph(s) that define the base
• Determine any points of intersection and/or zeros
• Sketch a representative cross section: pay attention to direction
• If shapes are vertical, you will use dx
• If shapes are horizontal, you will use dy
• Re-write equations and find your bounds in terms of x or y
(match the dx or dy)
• Write an equation for the length of the line that determines the
bottom of each shape
• Write an equation for the area of the shape
b
• Integral: A =
area of shape dx (or dy!)
a
• Solve the Integral (use your calculator unless it is a very simple
integral)
ò
The base of a solid is bounded by y = x and x = 9. The cross
sections of the solid taken perpendicular to the x-axis are
squares.
The base of a solid is enclosed by y = x, y = 5, and x = 0 and the
cross sections are semicircles, perpendicular to the x-axis. Find
the volume of the solid.
The base of a solid is the region enclosed by a circle centered at
the origin with a radius of 5 inches. Find the volume of the solid
if all cross sections perpendicular to the x-axis are squares.
A mathematician has a paperweight made so that its base is the
shape of the region between the x-axis and one arch of the
curve y = 2sinx. Each cross-section perpendicular to the x-axis is
a semicircle whose diameter runs from the x-axis to the curve.
Find the volume of the paperweight.
The base of a solid is the region in the first quadrant bounded by
the graph of y = 3x1/2 – x3/2 and the x-axis. Cross sections
perpendicular to the x-axis are isosceles right triangles, with one
leg on the xy-plane. What is the volume of the solid?
The base of a solid is the elliptical region with boundary curve
9x2 + 4y2 = 36. Cross sections perpendicular to the x-axis are
rectangles with height 3. Find the volume of the solid.
The base of a solid is a region bounded by the curves y = x2 and
y = 1. Cross sections perpendicular to the y-axis are semicircles.
Find the volume of the solid.
The base of a solid is the region bounded by the graphs of
x2 = 16y and y = 2. Cross sections perpendicular to the x-axis are
rectangles whose heights are half the size of the base.