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Volumes of Revolution

Spin a region around an axis and it sweeps out a solid. Slice it into disks or washers, add up their volumes, and the sum becomes an integral.

Cone · step 1 of 31

Side view: the region (y = x/2 from 0 to 3) and the slices

Speed

Take the region under y = x/2 from x = 0 to 3 — a right triangle — and spin it around the x-axis. It sweeps out a cone. How much space does it fill?

R(x) = \tfrac{x}{2}

V = \int_{0}^{3} \pi R(x)^2\,dx = \;?

Spin a region around an axis and it sweeps out a solid. Slice it into disks or washers, add up their volumes, and the sum becomes an integral.

Cone · step 1 of 31

Side view: the region (y = x/2 from 0 to 3) and the slices

Speed

Take the region under y = x/2 from x = 0 to 3 — a right triangle — and spin it around the x-axis. It sweeps out a cone. How much space does it fill?

R(x) = \tfrac{x}{2}

V = \int_{0}^{3} \pi R(x)^2\,dx = \;?