# How do you find the volume of the solid generated by revolving the region bounded by the graphs #y=x^2, y=4x-x^2#, about the x-axis, the line y=3?

See the answer below:

Credits:

Thanks to timwaite123 who reminded us of how to construct a quadratic equation given the coordinates of the vertex and the two-point coordinates in the curve.

By signing up, you agree to our Terms of Service and Privacy Policy

To find the volume of the solid generated by revolving the region bounded by the graphs (y=x^2), (y=4x-x^2), about the x-axis, and the line (y=3), we can use the method of cylindrical shells.

The volume (V) of the solid can be calculated using the formula:

[V = 2\pi \int_{a}^{b} (y \cdot r) ,dx]

where (r) represents the distance from the axis of revolution to the outer edge of the region, and (a) and (b) are the x-values where the two curves intersect.

First, we find the points of intersection by setting the equations (y=x^2) and (y=4x-x^2) equal to each other:

[x^2 = 4x - x^2]

[2x^2 - 4x = 0]

[2x(x - 2) = 0]

This gives us (x = 0) and (x = 2).

Next, we determine the radius (r), which is the distance from the axis of revolution (the line (y=3)) to the curve. The radius is given by (r = 3 - y).

Now, we integrate:

[V = 2\pi \int_{0}^{2} (3 - y) \cdot ((4x - x^2) - x^2) ,dx]

After simplifying, we integrate from (x = 0) to (x = 2):

[V = 2\pi \int_{0}^{2} (3 - x^2 - (4x - x^2)) ,dx]

[V = 2\pi \int_{0}^{2} (3 - 4x) ,dx]

[V = 2\pi \left[3x - 2x^2\right]_{0}^{2}]

[V = 2\pi \left[(3 \cdot 2 - 2 \cdot 2^2) - (3 \cdot 0 - 2 \cdot 0^2)\right]]

[V = 2\pi \left[6 - 8\right]]

[V = 2\pi (-2)]

[V = -4\pi]

Therefore, the volume of the solid generated by revolving the region about the x-axis and the line (y=3) is (4\pi) cubic units.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the general solution of the differential equation? : #(ylny)dx-xdy=0 #
- What is the arclength of #f(x)=sqrt((x+3)(x/2-1))+5x# on #x in [6,7]#?
- What is the integral of the momentum function?
- What is the arclength of #f(x)=e^(1/x)/x# on #x in [1,2]#?
- How do you find the average value of the function for #f(x)=x^3, 0<=x<=2#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7