The region is bounded by the given curves #y=x, y=4-x, 0

Question

The region is bounded by the given curves #y=x, y=4-x, 0<=x<=2# is rotated about the x-axis, how do you find the volume of the two solids of revolution?

Cameron Alexander · Accepted Answer

See below.

I am reading this as the volume of the shaded area A rotated around the x axis, and the volume of shaded area B rotated around the x axis. First we find the volume of B, then we find the volume of A+B and subtract volume B from this to find the volume A

Volume of B

#V=pi int_(0)^(2)(x)^2 dx=1/3x^3#

#V=pi[1/3x^3]_(0)^(2)=[1/3(2)^3]-[1/3(0)^3]=(8pi)/3#

Volume of A + B

#(4-x)^2=16-8x+x^2#

#V=pi int_(0)^(2)(16-8x+x^2) dx=16x-4x^2+1/3x^3#

#->=[16x-4x^2+1/3x^3]_(0)^(2)#

#=[16(2)-4(2)^2+1/3(2)^3]-[16(0)-4(0)^2+1/3(0)^3]#

#V=pi[56/3]=(56pi)/3#

Volume of A = (A + B) - B.

#V=(56pi)/3-(8pi)/3=16pi#

So:

Volume of A = #color(blue)(16pi)# cubic units.

Volume of B = #color(blue)((8pi)/3)# cubic units.

Revolution of A:

Revolution of B:

Luke Alvarez · Answer

undefined To find the volume of the two solids of revolution, you first need to find the points of intersection of the curves y = x and y = 4 - x. These points are (2, 2) and (0, 4).

For the first solid, the outer curve is y = 4 - x and the inner curve is y = x.

For the second solid, the outer curve is y = x and the inner curve is y = 4 - x.

Using the disk method, the volume of each solid can be calculated by integrating the area of the cross-sections perpendicular to the x-axis.

For the first solid:
$$V_1 = \pi \int_0^2 (4-x)^2 - x^2 \,dx$$

For the second solid:
$$V_2 = \pi \int_0^2 x^2 - (4-x)^2 \,dx$$

Integrate each expression over the given interval to find the volume of each solid.

The region is bounded by the given curves #y=x, y=4-x, 0&lt;=x&lt;=2# is rotated about the x-axis, how do you find the volume of the two solids of revolution?

The region is bounded by the given curves #y=x, y=4-x, 0<=x<=2# is rotated about the x-axis, how do you find the volume of the two solids of revolution?