$\int_{c}^{d} \pi (f(x) - b)^2 dx$

$\int_{c}^{d} \pi (f(y) - a)^2 dy$

$\pi r^2$

$r = f(x) - b$

$r = f(y) - a$

$\int_{a}^{b} A(x) dx$ or $\int_{a}^{b} A(y) dy$, where A is the area of the cross-section.

Solve $f(x) = g(x)$

$\int a f(x) dx = a \int f(x) dx$

$\int_{a}^{b} \pi [f(x)]^2 dx$

$\int_{c}^{d} \pi [g(y)]^2 dy$

A technique to find the volume of a solid of revolution by summing the volumes of thin discs.

A 3D shape formed by rotating a 2D region around an axis.

The distance between the function and the axis of rotation.

Integration sums the volumes of infinitely many thin discs to find the total volume.

A circle perpendicular to the axis of rotation.

A horizontal line around which a 2D region is rotated to create a solid.

A vertical line around which a 2D region is rotated to create a solid.

The space occupied by a circle, calculated as $\pi r^2$.

The limits on the integral that define the interval over which the volume is calculated.

$\pi$ is used to calculate the area of the circular cross-sections in the disc method.

The disc method calculates volume by summing the areas of infinitely thin circular discs perpendicular to the axis of rotation. Each disc's volume is $\pi r^2 * thickness$, and integration sums these volumes.

If rotating around a horizontal axis (y=b), integrate with respect to x. If rotating around a vertical axis (x=a), integrate with respect to y.

The radius 'r' represents the distance from the function to the axis of rotation and determines the area of each disc.

Visualization helps determine the correct radius and bounds of integration, ensuring the integral accurately represents the volume.

The limits of integration are the x-values (for horizontal axis) or y-values (for vertical axis) where the region begins and ends, often found by intersection points.

The disc method extends the concept of area to volume by summing infinitesimally thin circular areas (discs) to create a 3D solid.

The radius changes to reflect the new distance between the function and the new axis of rotation. It becomes the absolute value of the function minus the axis value.

We square the radius because the area of a circle is $\pi r^2$, and we are summing the areas of these circular cross-sections to find the volume.

Integrate $\pi [f(x)]^2$ from a to b with respect to x, where f(x) is the function defining the region and a and b are the limits of integration.

Integrate $\pi [g(y)]^2$ from c to d with respect to y, where g(y) is the function defining the region and c and d are the limits of integration.