Volumes of Revolution Math Example 4

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

Example 4

medium
Use the shell method to find the volume when the region bounded by y=x2y = x^2, x=0x=0, y=1y=1 is rotated around the yy-axis.

Solution

  1. 1
    Shell method: V=2ฯ€โˆซ01xโ‹…(1โˆ’x2)โ€‰dx=2ฯ€โˆซ01(xโˆ’x3)โ€‰dxV = 2\pi\int_0^1 x \cdot (1-x^2)\,dx = 2\pi\int_0^1(x-x^3)\,dx.
  2. 2
    =2ฯ€[x22โˆ’x44]01=2ฯ€(12โˆ’14)=ฯ€2= 2\pi\left[\frac{x^2}{2}-\frac{x^4}{4}\right]_0^1 = 2\pi(\frac{1}{2}-\frac{1}{4}) = \frac{\pi}{2}.

Answer

ฯ€2\frac{\pi}{2}
The shell height is the vertical extent of the region at each xx, which is 1โˆ’x21 - x^2. Shells integrate parallel to the axis of rotation.

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 2 hard

Find the volume when the region between [formula] and [formula] ([formula]) is rotated around the [f

Example 3 easy

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

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