Math · Calculus II · Worked example

Find a volume with the disk method

The region under y = √x from x = 0 to x = 4 is revolved about the x-axis. Find the volume of the solid.

y = \sqrt{x}

Picture a slice

A slice perpendicular to the x-axis at x is a disk. Its radius is the height of the curve there, √x.

Write the area of a slice

Squaring the radius removes the root.

A (x) = π {(\sqrt{x})}^{2} = π x

Integrate the slices

Add the disks from x = 0 to x = 4.

V = \int_{0}^{4} π x d x = π \cdot \frac{4^{2}}{2} = 8 π

Interpret

8π ≈ 25.1 cubic units. The solid is a bowl-shaped paraboloid lying on its side, 4 units long with an opening of radius 2.

Result

V = 8π ≈ 25.1 cubic units.

Your turn

Revolve the region under y = x² from x = 0 to x = 1 about the x-axis. Find the volume.

Show the answer and explanation

π/5.

Each slice is a disk of radius x², so its area is π(x²)² = πx⁴, and V = ∫₀¹ πx⁴ dx = π/5.

\begin{array}{l} y = x^{2} \\ \int_{0}^{1} π x^{4} d x = π \cdot \frac{1^{5}}{5} = \frac{π}{5} \end{array}

Keep exploring

In Slopes, sums & signed area, the integrand πx on [0, 4] has signed area 25.13, which is 8π. Change the upper limit to 2: the volume falls to 2π, a quarter rather than a half, because the largest disks are near x = 4.

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