Volumes of Revolution Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Find the volume when the region between y=xy=x and y=x2y=x^2 (0โ‰คxโ‰ค10 \leq x \leq 1) is rotated around the xx-axis.

Solution

  1. 1
    On [0,1][0,1]: xโ‰ฅx2x \geq x^2, so outer radius R=xR=x, inner radius r=x2r=x^2.
  2. 2
    Washer formula: V=ฯ€โˆซ01(x2โˆ’x4)โ€‰dxV = \pi\int_0^1(x^2-x^4)\,dx.
  3. 3
    =ฯ€[x33โˆ’x55]01=ฯ€(13โˆ’15)=2ฯ€15= \pi\left[\frac{x^3}{3}-\frac{x^5}{5}\right]_0^1 = \pi\left(\frac{1}{3}-\frac{1}{5}\right) = \frac{2\pi}{15}.

Answer

2ฯ€15\frac{2\pi}{15}
The washer method subtracts the inner radius squared from the outer radius squared. Remember: R2โˆ’r2R^2 - r^2, not (Rโˆ’r)2(R-r)^2.

About Volumes of Revolution

Finding the volume of a three-dimensional solid formed by rotating a two-dimensional region around an axis. The disc/washer method uses circular cross-sections perpendicular to the axis; the shell method uses cylindrical shells parallel to the axis.

Learn more about Volumes of Revolution โ†’

More Volumes of Revolution Examples

Example 1 easy

Find the volume of the solid formed by rotating [formula] around the [formula]-axis from [formula] t

Example 3 easy

Find the volume of the solid formed by rotating [formula] from [formula] to [formula] around the [fo

Example 4 medium

Use the shell method to find the volume when the region bounded by [formula], [formula], [formula] i

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