Find the surface area formed by revolving the graph of f(x)=81-x^2 on the interval [0,9] about the y-axis. (express the answer in terms of pie.)
To find the surface area generated by revolving the graph of a function around the y-axis, we can use the method of cylindrical shells.
The formula for the surface area using cylindrical shells is given by:
Surface Area = 2π ∫(x * height) dx
In this case, we need to find the height and x-value of each cylindrical shell.
First, let's find the height of each cylindrical shell. The height of each shell will vary depending on the x-value on the graph. Since we are revolving the graph around the y-axis, the height will be the distance from the x-axis to the function f(x) at each x-value.
In this case, the height will be given by f(x) = 81 - x^2.
Next, we need to find the x-value range for integration. The interval [0, 9] is the range for x-values. Therefore, the integral will be taken from x = 0 to x = 9.
Now, we can set up the integral to find the surface area:
Surface Area = 2π ∫(x * height) dx
Substituting the height as f(x) = 81 - x^2:
Surface Area = 2π ∫(x * (81 - x^2)) dx
Integrating the expression gives:
Surface Area = 2π ∫(81x - x^3) dx
Evaluating the integral from x = 0 to x = 9:
Surface Area = 2π [ (81/2)x^2 - (1/4)x^4 ] evaluated from 0 to 9
Surface Area = 2π [ (81/2)(9)^2 - (1/4)(9)^4 ] - 2π [ (81/2)(0)^2 - (1/4)(0)^4 ]
Simplifying further:
Surface Area = 2π [ (81/2)(81) - 0 ] - 2π [ 0 - 0 ]
Surface Area = 2π [ 6561/2 ]
Finally, expressing the answer in terms of π:
Surface Area = 6561π
Therefore, the surface area formed by revolving the graph of f(x) = 81 - x^2 on the interval [0, 9] about the y-axis is 6561π.