8.12 Volume with Washer Method: Revolving Around Other Axes

The Washer Method is a technique used to find the volume of a solid formed by revolving a region around an axis other than the x- or y-axis. The method involves slicing the solid into thin washers and finding the volume of each washer. The total volume of the solid is found by summing the volumes of all the washers. Here are the steps to use the Washer Method: Identify the region being revolved: The region should be defined by two functions, f(x), g(y), h(x), or k(y), and should be bounded by lines, x = a and x = b or y = c and y = d. The two functions define the inner and outer radii of each washer. Choose the axis of revolution: If the region is being revolved around the x-axis, the width of each washer will be dx and the inner and outer radii will be f(x) and h(x), respectively. If the region is being revolved around the y-axis, the width of each washer will be dy and the inner and outer radii will be g(y) and k(y), respectively. If the region is being revolved around an axis other than the x- or y-axis, the width of each washer will be the differential element, dA. Find the volume of each washer: To find the volume of each washer, subtract the volume of the smaller disk from the volume of the larger disk. The volume of each disk is found by multiplying π by the square of the radius and the width of the disk. Integrate the function for the volume of each washer: To find the total volume of the solid, integrate the function for the volume of each washer with respect to x, y, or the differential element, dA.

Example 1: Consider a region defined by the functions f(x) = x^2 and h(x) = x + 1, revolved around the z-axis from x = 0 to x = 1. The width of each washer is dA and the inner and outer radii are f(x) = x^2 and h(x) = x + 1, respectively. The volume of each washer is found by subtracting the volume of the smaller disk from the volume of the larger disk. This gives us π(h^2(x) - f^2(x)) * dA. To find the total volume of the solid, we integrate this function from x = 0 to x = 1. This gives us the definite integral from 0 to 1 of π(h^2(x) - f^2(x)) dx = (3π/4). Example 2: Consider a region defined by the functions g(y) = √(4-y^2) and k(y) = √(9-y^2), revolved around the w-axis from y = 0 to y = 2. The width of each washer is dA and the inner and outer radii are g(y) = √(4-y^2) and k(y) = √(9-y^2), respectively. The volume of each washer is found by subtracting the volume of the smaller disk from the volume of the larger disk. This gives us π(k^2(y) - g^2(y)) * dA. To find the total volume of the solid, we integrate this function from y = 0 to y = 2. This gives us the definite integral from 0 to 2 of π(k^2(y) - g^2(y)) dy = (15π/8).

Example 3: Consider a region defined by the functions f(x) = x^2 and h(x) = 3x, revolved around the z-axis from x = 1 to x = 2. The width of each washer is dA and the inner and outer radii are f(x) = x^2 and h(x) = 3x, respectively. The volume of each washer can be found using the formula π(h^2(x) - f^2(x)) * dA. To find the total volume of the solid, we integrate this function from x = 1 to x = 2. This gives us the definite integral from 1 to 2 of π(h^2(x) - f^2(x)) dx = (25π/3).

Example 4: Consider a region defined by the functions g(y) = 2 and k(y) = 4, revolved around the w-axis from y = 0 to y = 1. The width of each washer is dA and the inner and outer radii are g(y) = 2 and k(y) = 4, respectively. The volume of each washer can be found using the formula π(k^2(y) - g^2(y)) * dA. To find the total volume of the solid, we integrate this function from y = 0 to y = 1. This gives us the definite integral from 0 to 1 of π(k^2(y) - g^2(y)) dy = (2π).

In conclusion, the Washer Method is a useful technique for finding the volume of a solid formed by revolving a region around an axis other than the x- or y-axis. The method involves slicing the solid into thin washers and finding the volume of each washer by subtracting the volume of the smaller disk from the volume of the larger disk. The total volume of the solid is found by integrating the function for the volume of each washer. The key to using the Washer Method is to correctly identify the region being revolved, choose the axis of revolution, and find the volume of each washer. With these steps, you can use the Washer Method to find the volume of complex solid shapes.