The region bounded by y=2.5x^2 and y=4x is to be rotated about both axes and the volume generated calculated by both the washer and the shell methods.
1)The volume of the region bounded by y=2.5x^2 and y=4x, when rotated about the x-axis is?
2) The volume of the region bounded by y=2.5x^2 and y=4x when rotated about the y-axis is?
Thank you
To find the volumes using the washer and shell methods, we first need to determine the limits of integration. Let's start by plotting the two given curves, y = 2.5x^2 and y = 4x:
1) The volume of the region bounded by y = 2.5x^2 and y = 4x, when rotated about the x-axis:
To find the limits of integration, we need to determine where the two curves intersect. Set them equal to each other:
2.5x^2 = 4x
Simplifying this equation gives:
2.5x^2 - 4x = 0
x(2.5x - 4) = 0
This equation has two solutions: x = 0 and x = 4/2.5 = 1.6. So the limits of integration are x = 0 to x = 1.6.
Now, we can calculate the volume using the washer method. The general formula for the volume of a washer is:
V = ∫ [π(R^2 - r^2)] dx
In this case, the outer radius (R) is given by y = 4x, and the inner radius (r) is given by y = 2.5x^2. So we have:
R = 4x
r = 2.5x^2
The volume is then:
V = ∫ [π(4x)^2 - π(2.5x^2)^2] dx
V = ∫ [16πx^2 - 6.25πx^4] dx
Integrating this expression with the limits of integration, x = 0 to x = 1.6, will give you the volume of the region bounded by y = 2.5x^2 and y = 4x, when rotated about the x-axis.
2) The volume of the region bounded by y = 2.5x^2 and y = 4x, when rotated about the y-axis:
Similarly, we need to find the limits of integration by setting the two curves equal to each other:
2.5x^2 = 4x
Simplifying this equation gives:
2.5x^2 - 4x = 0
x(2.5x - 4) = 0
Solving for x, we find x = 0 and x = 4/2.5 = 1.6. These are the limits of integration.
To use the shell method, the general formula for volume is:
V = ∫ [2πx(f(x) - g(x))] dx
In this case, f(x) is given by y = 4x, and g(x) is given by y = 2.5x^2. So we have:
f(x) = 4x
g(x) = 2.5x^2
The volume is then:
V = ∫ [2πx(4x - 2.5x^2)] dx
Integrating this expression with the limits of integration, x = 0 to x = 1.6, will give you the volume of the region bounded by y = 2.5x^2 and y = 4x, when rotated about the y-axis.
Please note that to obtain the exact numerical values for the volumes, you need to evaluate the integrals using calculus techniques.