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?
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 to do it:
1. First, let's find the points of intersection between the two curves. Set f(x) equal to g(x) and solve for x:
x^2 = 4x
Rearrange the equation:
x^2 - 4x = 0
Factor:
x(x - 4) = 0
So, we have two solutions: x = 0 and x = 4. These are the x-coordinates where the curves intersect.
2. Now, let's set up the integral to calculate the volume using cylindrical shells. The volume of an individual cylindrical shell can be calculated as follows:
V = 2πrhΔx
- We choose the axis of rotation as y = 20, so we'll be integrating with respect to y.
- The radius of each cylindrical shell will be the distance from the axis of rotation (y = 20) to the curve f(x).
- The height (h) of each cylindrical shell will be the difference in x-coordinates between the two curves: Δx = x - 4x.
3. To calculate the radius, we need to express f(x) in terms of y. Rearrange the equation f(x) = x^2 to solve for x in terms of y:
x = √y
The radius of each cylindrical shell is y - 20.
4. To calculate the height, we use the difference between the x-coordinates of the curves. Δx = x - 4x = √y - 4√y = -3√y.
5. Finally, the integral for the volume becomes:
V = ∫[a,b] 2π(rh) dy
= ∫[0,16] 2π(y - 20)(-3√y) dy
Integrate the expression between 0 and 16 to find the volume.
Note: The integration limits [a,b] are determined by the y-values where the curves intersect. In this case, the area enclosed between the curves is from y = 0 to y = 16.
By evaluating the integral, you should be able to find the volume obtained by revolving the area between the curves around the line y = 20.