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AT.G.2.c

Use Cavalieri’s Principle to determine volume of three-dimensional figures.

Example Problems

The base of a solid is the region enclosed by the graphs of

y = \ln(x)

and

y = 0.1x^3 - 0.5x^2

, between

x = 2

and

x = 5

.

Cross sections of the solid perpendicular to the x-axis are rectangles whose height is

6 - x

.

What definite integral gives the volume of the solid?

The base of a solid is the region enclosed by the graphs of

y = \ln(x + 2)

and

y = \frac{x}{4}

, between the y-axis and

x = 4

.

Cross sections of the solid perpendicular to the x-axis are rectangles whose height is

5 - x

.

What definite integral gives the volume of the solid?

The base of a solid is the region enclosed by the graphs of

y = \sin(x)

and

y = 4 - \sqrt{x}

, between

x = 2

and

x = 7

.

Cross sections of the solid perpendicular to the x-axis are rectangles whose height is

2x

.

What definite integral gives the volume of the solid?

Khan Academy Resources

Cavalieri's principle in 3D Apply Cavalieri's principle Cavalieri's principle in 3D Volumes of cones intuition

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