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  2. AP Calculus
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Steps to find volume using the Washer Method.

  1. Sketch the region. 2. Identify the axis of revolution. 3. Determine the outer radius R(x) and inner radius r(x). 4. Find the limits of integration. 5. Set up and evaluate the integral: ∫abπ[R(x)2−r(x)2]dx\int_{a}^{b} \pi [R(x)^2 - r(x)^2] dx∫ab​π[R(x)2−r(x)2]dx.
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Steps to find volume using the Washer Method.

  1. Sketch the region. 2. Identify the axis of revolution. 3. Determine the outer radius R(x) and inner radius r(x). 4. Find the limits of integration. 5. Set up and evaluate the integral: ∫abπ[R(x)2−r(x)2]dx\int_{a}^{b} \pi [R(x)^2 - r(x)^2] dx∫ab​π[R(x)2−r(x)2]dx.

How to determine the outer and inner radii?

Identify the function farther from the axis of revolution (outer radius) and the function closer to the axis (inner radius).

Steps to find the intersection points of two functions.

  1. Set the two functions equal to each other. 2. Solve for x (or y). 3. These x (or y) values are the limits of integration.

How to handle revolving around a line other than the x or y-axis?

Adjust the radius expressions by subtracting the value of the axis of revolution from the function: R(x) = f(x) - b.

How to choose between integrating with respect to x or y?

If the axis of revolution is horizontal, integrate with respect to x. If the axis of revolution is vertical, integrate with respect to y.

How to simplify the integral after setting it up?

Expand the squared terms, combine like terms, and simplify the expression before integrating.

What should you do if you can't find intersection points algebraically?

Use a graphing calculator to find the approximate intersection points.

How to check if your answer is reasonable?

Estimate the volume using geometric shapes and compare it to your calculated volume.

What is the first step when encountering a volume problem?

Draw a diagram of the region and the axis of revolution.

How do you deal with absolute values when calculating radii?

Ensure that the radius is always positive by using absolute values or by carefully considering which function is further from the axis of revolution.

Disc Method vs. Washer Method.

Disc: Solid has no hole, single radius. Washer: Solid has a hole, inner and outer radii.

Revolving around x-axis vs. revolving around y-axis.

x-axis: Integrate with respect to x, radii are vertical. y-axis: Integrate with respect to y, radii are horizontal.

Revolving around y=b vs. revolving around x=a.

y=b: Horizontal axis, radii are vertical, functions in terms of x. x=a: Vertical axis, radii are horizontal, functions in terms of y.

Washer Method vs. Shell Method.

Washer: Integrate perpendicular to the axis, uses radii. Shell: Integrate parallel to the axis, uses height and radius.

Outer radius vs. Inner radius.

Outer: Distance from axis to the farthest curve. Inner: Distance from axis to the closest curve.

Integration with respect to x vs. Integration with respect to y.

x: Used for horizontal axes, functions in terms of x. y: Used for vertical axes, functions in terms of y.

Washer Method with horizontal axis vs. Washer Method with vertical axis.

Horizontal: V=∫π[(f(x)−b)2−(g(x)−b)2]dxV = \int \pi [(f(x)-b)^2 - (g(x)-b)^2] dxV=∫π[(f(x)−b)2−(g(x)−b)2]dx. Vertical: V=∫π[(f(y)−a)2−(g(y)−a)2]dyV = \int \pi [(f(y)-a)^2 - (g(y)-a)^2] dyV=∫π[(f(y)−a)2−(g(y)−a)2]dy.

Choosing between Disc/Washer and Shell Method.

Disc/Washer: Easier when axis is perpendicular to the slicing. Shell: Easier when axis is parallel to the slicing.

Washer Method with simple functions vs. complex functions.

Simple: Easier to find intersection points and integrate. Complex: May require calculator for intersection and more advanced integration techniques.

Solids with constant cross-sections vs. variable cross-sections.

Constant: Volume = Area * Height. Variable: Volume = Integral of Area function over the interval.

Washer Method formula revolving around a horizontal line y=by=by=b.

∫cdπ[(f(x)−b)2−(g(x)−b)2]dx\int_{c}^{d}\pi [(f(x)-b)^2 - (g(x)-b)^2] dx∫cd​π[(f(x)−b)2−(g(x)−b)2]dx

Washer Method formula revolving around a vertical line x=ax=ax=a.

∫cdπ[(f(y)−a)2−(g(y)−a)2]dy\int_{c}^{d}\pi [(f(y)-a)^2 - (g(y)-a)^2] dy∫cd​π[(f(y)−a)2−(g(y)−a)2]dy

General formula for the Washer Method.

∫cdπ(R(x)2−r(x)2)dx\int_{c}^{d}\pi (R(x)^2 - r(x)^2) dx∫cd​π(R(x)2−r(x)2)dx, where R(x) is the outer radius and r(x) is the inner radius.

Area of a circle.

A=πr2A = \pi r^2A=πr2

Volume of a solid of revolution using cross-sections.

V=∫abA(x)dxV = \int_{a}^{b} A(x) dxV=∫ab​A(x)dx, where A(x) is the area of the cross-section at x.

How to calculate the outer radius when revolving around y = b?

R(x)=∣f(x)−b∣R(x) = |f(x) - b|R(x)=∣f(x)−b∣, where f(x) is the function farther from the axis of revolution.

How to calculate the inner radius when revolving around y = b?

r(x)=∣g(x)−b∣r(x) = |g(x) - b|r(x)=∣g(x)−b∣, where g(x) is the function closer to the axis of revolution.

What is the general form of an integral for volume of revolution?

∫abπ(radius)2dx\int_{a}^{b} \pi (radius)^2 dx∫ab​π(radius)2dx

How do you determine the limits of integration?

Find the points of intersection between the curves, these x or y values will be your limits.

What's the formula for volume using the Washer Method with respect to y?

V=∫cdπ[(R(y))2−(r(y))2]dyV = \int_{c}^{d} \pi [(R(y))^2 - (r(y))^2] dyV=∫cd​π[(R(y))2−(r(y))2]dy