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Formula for volume with disc method (horizontal axis of rotation y=b)?

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

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Formula for volume with disc method (horizontal axis of rotation y=b)?
$\int_{c}^{d} \pi (f(x) - b)^2 dx$
Formula for volume with disc method (vertical axis of rotation x=a)?
$\int_{c}^{d} \pi (f(y) - a)^2 dy$
Area of a circle?
$\pi r^2$
How to determine the radius when rotating around y=b?
$r = f(x) - b$
How to determine the radius when rotating around x=a?
$r = f(y) - a$
What is the general form of an integral for volume?
$\int_{a}^{b} A(x) dx$ or $\int_{a}^{b} A(y) dy$, where A is the area of the cross-section.
Formula for finding intersection points of two functions f(x) and g(x)?
Solve $f(x) = g(x)$
What is the formula for the constant multiple rule in integration?
$\int a f(x) dx = a \int f(x) dx$
What is the general form of the volume integral when rotating around the x-axis?
$\int_{a}^{b} \pi [f(x)]^2 dx$
What is the general form of the volume integral when rotating around the y-axis?
$\int_{c}^{d} \pi [g(y)]^2 dy$
What is the disc method?
A technique to find the volume of a solid of revolution by summing the volumes of thin discs.
What is a solid of revolution?
A 3D shape formed by rotating a 2D region around an axis.
Define the radius in the disc method.
The distance between the function and the axis of rotation.
What is the role of integration in the disc method?
Integration sums the volumes of infinitely many thin discs to find the total volume.
What is the cross-section in the disc method?
A circle perpendicular to the axis of rotation.
What is the horizontal axis of rotation?
A horizontal line around which a 2D region is rotated to create a solid.
What is the vertical axis of rotation?
A vertical line around which a 2D region is rotated to create a solid.
What is the area of a circle?
The space occupied by a circle, calculated as $\pi r^2$.
Define the term 'bounds of integration'.
The limits on the integral that define the interval over which the volume is calculated.
What is the role of $\pi$ in volume calculation?
$\pi$ is used to calculate the area of the circular cross-sections in the disc method.
Explain how the disc method works.
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.
How does the axis of rotation affect the integral setup?
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.
What is the significance of the radius 'r' in the disc method?
The radius 'r' represents the distance from the function to the axis of rotation and determines the area of each disc.
Why is it important to visualize the solid of revolution?
Visualization helps determine the correct radius and bounds of integration, ensuring the integral accurately represents the volume.
How do you determine the limits of integration?
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.
Explain the relationship between area and volume in the disc method.
The disc method extends the concept of area to volume by summing infinitesimally thin circular areas (discs) to create a 3D solid.
What happens to the radius when the axis of rotation changes?
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.
Why do we square the radius in the disc method?
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.
Explain how to find the volume of a solid of revolution when rotating around the x-axis.
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.
Explain how to find the volume of a solid of revolution when rotating around the y-axis.
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.