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

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

$\pi r^2$

$r = f(x) - b$

$r = f(y) - a$

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

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

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

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

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

1. Identify f(x) and b. 2. Determine the limits of integration (c, d). 3. Set up the integral: $\int_{c}^{d} \pi (f(x) - b)^2 dx$. 4. Evaluate the integral.

1. Identify f(y) and a. 2. Determine the limits of integration (c, d). 3. Set up the integral: $\int_{c}^{d} \pi (f(y) - a)^2 dy$. 4. Evaluate the integral.

1. Sketch the region. 2. Identify the axis of rotation. 3. Determine the radius. 4. Set up the integral. 5. Evaluate the integral.

If rotating around a horizontal line, use f(x). If rotating around a vertical line, use f(y).

1. Set f(x) = g(x). 2. Solve for x. 3. The solutions are the x-coordinates of the intersection points.

Remember to subtract the negative value correctly, which results in adding the absolute value of 'b' to f(x) in the integral.

Take the derivative of your solution and see if it matches the original integrand.

Evaluate the integral using appropriate integration techniques (u-substitution, etc.) and then apply the limits of integration.

Use techniques like u-substitution, integration by parts, or trigonometric identities to simplify the integral before evaluating.

Use numerical methods or a calculator to approximate the value of the definite integral.

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.