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How to find the volume when rotating around y=b?
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.
How to find the volume when rotating around x=a?
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.
Steps to solve volume problems with the disc method?
1. Sketch the region. 2. Identify the axis of rotation. 3. Determine the radius. 4. Set up the integral. 5. Evaluate the integral.
How to determine the function to use, f(x) or f(y)?
If rotating around a horizontal line, use f(x). If rotating around a vertical line, use f(y).
How to find the intersection points of f(x) and g(x)?
1. Set f(x) = g(x). 2. Solve for x. 3. The solutions are the x-coordinates of the intersection points.
How do you handle a negative value for 'b' in $y=b$?
Remember to subtract the negative value correctly, which results in adding the absolute value of 'b' to f(x) in the integral.
How to check if your integration is correct?
Take the derivative of your solution and see if it matches the original integrand.
What do you do after setting up the integral?
Evaluate the integral using appropriate integration techniques (u-substitution, etc.) and then apply the limits of integration.
How do you handle complex integrals in volume problems?
Use techniques like u-substitution, integration by parts, or trigonometric identities to simplify the integral before evaluating.
What should you do if you cannot find an elementary antiderivative?
Use numerical methods or a calculator to approximate the value of the definite integral.
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.