consider the area enclosed between the curves f (x) = x2 and g (x) = 4x
what is the volume obtained by revolving the area between these two curves around the line y = 20 ?
please help, i don't know what to do
To find the volume obtained by revolving the area between the curves around the line y = 20, we can use the method of cylindrical shells. Here's how you can approach this problem:
1. First, we need to identify the bounds of integration. To do this, find the x-values where the two curves intersect by setting f(x) equal to g(x) and solving for x:
x^2 = 4x
x^2 - 4x = 0
x(x - 4) = 0
The solutions are x = 0 and x = 4. So the bounds of integration will be from x = 0 to x = 4.
2. Next, we need to express the height of each cylindrical shell as a function of x. In this case, the height of each shell will be the difference between the line y = 20 and the function g(x):
h(x) = 20 - g(x)
= 20 - 4x
3. Determine the radius of each cylindrical shell. The radius will be the x-value, as we are revolving around the line y = 20.
4. Now, we can calculate the volume of each cylindrical shell. The volume of a cylindrical shell is given by the formula:
V(x) = 2πr(x) * h(x) * Δx
Where Δx is an infinitesimally small change in x. In this case, we can approximate Δx as dx, which will represent an infinitesimally small width of each shell.
5. Integrate the expression V(x) with respect to x over the bounds of integration:
Volume = ∫[0 to 4] 2πx * (20 - 4x) dx
6. Evaluate the integral using techniques such as integration by parts or the inverse power rule.
Once you perform the integration, you'll find the value of the volume obtained by revolving the area between the curves around the line y = 20.