Answer :
To find the volume of the solid, we need to integrate the area of each cross section perpendicular to the x-axis. Since each cross section is a square, the area of each cross section will be the side length squared.
To determine the side length of each square, we need to find the length of the base of the square at each x-coordinate. This can be done by finding the y-coordinate on the graph of y=4-2x.
Let's set up the integral to find the volume:
∫[a,b] (4-2x)^2 dx
To find the limits of integration, we need to determine the x-values where the graph of y=4-2x intersects the x-axis. Setting y=0, we get:
0 = 4-2x
2x = 4
x = 2
So, the limits of integration are from x=0 to x=2.
Now, let's evaluate the integral:
∫[0,2] (4-2x)^2 dx
= ∫[0,2] (16 - 16x + 4x^2) dx
= [16x - 8x^2 + (4/3)x^3] evaluated from x=0 to x=2
= (16(2) - 8(2)^2 + (4/3)(2)^3) - (16(0) - 8(0)^2 + (4/3)(0)^3)
= (32 - 32 + (32/3)) - (0 - 0 + 0)
= 32/3
Therefore, the volume of the solid is 32/3 cubic units.
So the answer is C. 32/3.
To determine the side length of each square, we need to find the length of the base of the square at each x-coordinate. This can be done by finding the y-coordinate on the graph of y=4-2x.
Let's set up the integral to find the volume:
∫[a,b] (4-2x)^2 dx
To find the limits of integration, we need to determine the x-values where the graph of y=4-2x intersects the x-axis. Setting y=0, we get:
0 = 4-2x
2x = 4
x = 2
So, the limits of integration are from x=0 to x=2.
Now, let's evaluate the integral:
∫[0,2] (4-2x)^2 dx
= ∫[0,2] (16 - 16x + 4x^2) dx
= [16x - 8x^2 + (4/3)x^3] evaluated from x=0 to x=2
= (16(2) - 8(2)^2 + (4/3)(2)^3) - (16(0) - 8(0)^2 + (4/3)(0)^3)
= (32 - 32 + (32/3)) - (0 - 0 + 0)
= 32/3
Therefore, the volume of the solid is 32/3 cubic units.
So the answer is C. 32/3.
