Volume of a solid of revolution

Area enclosed by a curve

The area between a curve y = f(x), the x-axis and the lines x = a and x = b is A = ∫ₐᵇ y dx (areas below the x-axis count as negative). The area between two curves is ∫ (upper − lower) dx.

Worked exampleArea under a curve

Find the area between y = x², the x-axis, x = 0 and x = 3.

Integrate.

Answer: The area is 9 square units.

The area from 0 to x is x³/3.

When the area under y = f(x) from x = a to x = b is rotated through 360° about the x-axis, it makes a solid of volume:

The volume of revolution.

Worked exampleA cone

The line y = x from x = 0 to x = 3 is rotated about the x-axis. Find the volume.

Use the formula.

Answer: The volume is 9π ≈ 28.3 cubic units, the volume of a cone with r = h = 3: ⅓π×9×3.

Length of an arc

The length of a curve from x = a to x = b.

For a straight line y = x from 0 to 3, dy/dx = 1, so s = ∫√2 dx = 3√2 ≈ 4.24, which is the length given by Pythagoras.

Try itArea and volume of y = x²

Choose the upper limit b. Area = b³/3 and volume of revolution about the x-axis = πb⁵/5.

Check yourself The area under y = 2x from 0 to 4 is

Check yourself The volume made by rotating y = x about the x-axis between 0 and 3 is

Check yourself An area below the x-axis gives an integral that is

Key terms

Solid of revolution: A solid made by rotating a curve about an axis.
Arc length: The length along a curve.
Definite integral: Integral between two limits.
Area between curves: The integral of the upper minus the lower curve.

Summary

Area = ∫ y dx between the limits.
Volume of revolution = π∫ y² dx.
Arc length = ∫ √(1 + (dy/dx)²) dx.
Between two curves integrate (upper − lower).