AP Calculus - Area, Volume, Arc Length

Finding the Area Between Two Curves Using Integration

When we have two curves, we can find the area between them by integrating the difference between the two curves. Here are the steps to follow:

1. Find the points of intersection between the two curves by setting them equal to each other and solving for x.

2. Determine which curve is on top and which is on bottom. This can be done by evaluating the y-values of each curve at the points of intersection found in step 1.

3. Set up the integral to find the area between the curves. If the top curve is f(x) and the bottom curve is g(x), then the integral is:

   ∫[a,b] (f(x) - g(x)) dx

   where a and b are the x-coordinates of the points of intersection found in step 1.

4. Integrate the function from step 3 with respect to x. This will give you the area between the two curves.

   Area = ∫[a,b] (f(x) - g(x)) dx

   Note that the area will be negative if the bottom curve is above the top curve.

5. Don't forget to include units for the area, which will be in square units.

That's it! By following these steps, you can find the area between two curves using integration.

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Finding the Volume of an Area Rotated Around the X or Y Axis

When finding the volume of an area rotated around the x or y axis, we use the method of cylindrical shells. This involves dividing the area into thin vertical strips, rotating each strip around the axis of rotation, and then adding up the volumes of all the resulting cylindrical shells.

Finding the Volume of an Area Rotated Around the X Axis

To find the volume of an area rotated around the x axis, we use the following formula:

V = ∫[a,b] 2πx * f(x) dx

where f(x) is the function that defines the area, and a and b are the limits of integration.

Finding the Volume of an Area Rotated Around the Y Axis

To find the volume of an area rotated around the y axis, we use the following formula:

V = ∫[c,d] 2πy * g(y) dy

where g(y) is the function that defines the area, and c and d are the limits of integration.

Example

Let's say we want to find the volume of the area bounded by the curves y = x^2 and y = 0, rotated around the x axis from x = 0 to x = 1. Using the formula for finding the volume of an area rotated around the x axis, we get:

V = ∫[0,1] 2πx * x^2 dx

Simplifying this integral, we get:

V = 2π/3

Therefore, the volume of the area rotated around the x axis is 2π/3 cubic units.

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Finding Arc Length using Integration

Arc length is the length of a curve in a plane. It can be found using integration. The formula for arc length is:

L = ∫[a,b] √(1 + (dy/dx)²) dx or L = ∫[c,d] √(1 + (dx/dy)²) dy

where a and b are the limits of integration for x and c and d are the limits of integration for y.

To find the arc length of a curve using integration, follow these steps:

1. Find the derivative of the function with respect to x or y.
2. Square the derivative and add 1.
3. Take the square root of the result.
4. Set up the integral with the limits of integration for x or y.
5. Integrate the function from step 3 with respect to x or y.

Example

Find the arc length of the curve y = x^(3/2) from x = 0 to x = 4.

1. Find the derivative of y with respect to x: dy/dx = (3/2)x^(1/2).
2. Square the derivative and add 1: 1 + (dy/dx)² = 1 + 9/4x.
3. Take the square root of the result: √(1 + (dy/dx)²) = √(1 + 9/4x).
4. Set up the integral: L = ∫[0,4] √(1 + 9/4x) dx.
5. Integrate the function: L = (8/27)(13√13 - 1).

Therefore, the arc length of the curve y = x^(3/2) from x = 0 to x = 4 is (8/27)(13√13 - 1).

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