The disk method finds the volume of a solid of revolution by integrating the cross-sectional area of circular disks perpendicular to the axis of rotation.
Formula:
$$ V = \pi \int_{a}^{b} \bigl[f(x)\bigr]^2 \, dx $$
Best used when: The region is bounded by \( y = f(x) \), \( y = 0 \), and vertical lines.
The washer method extends the disk method for regions with a hole.
Formula:
$$ V = \pi \int_{a}^{b} \Bigl([R(x)]^2 - [r(x)]^2\Bigr)\, dx $$
Best used when: The region is bounded by two functions and rotated about an axis.
The shell method calculates the volume by integrating thin cylindrical shells.
Formula:
$$ V = 2\pi \int_{a}^{b} x\,f(x)\,dx $$
Best used when: The region is rotated about an axis parallel to the y-axis.
This method decomposes the solid into nested cylindrical shells.
Formula:
$$ V = 2\pi \int_{a}^{b} y\,h(y)\,dy $$
Best used when: The region is defined in terms of y.
Calculate to see result
Calculate to see formula