Write the integral in one variable to find the volume of the solid obtained by rotating the first-quadrant region bounded by y = 0.5x2 and y = x about the line x = 7. (10 points)
The curves intersect at (0,0) and (2,2)
using shells of thickness dx,
v = ∫[0,2] 2πrh dx
where r = 7-x and h=x - 0.5x^2
v = ∫[0,2] 2π(7-x)(x - 0.5x^2) dx = 8π
using discs (washers) of thickness dy,
v = ∫[0,2] π(R^2-r^2) dy
where R = 7-y and r = 7-√(2y)
v = ∫[0,2] π((7-y)^2-(7-√(2y))^2) dy = 8π
slrtttt
To find the volume of the solid obtained by rotating the region bounded by y=0.5x^2 and y=x about the line x=7, we can use the method of cylindrical shells.
First, let's plot the region in the first quadrant bounded by y=0.5x^2 and y=x:
We see that the region is bounded by y=0.5x^2 (blue curve) and y=x (green line) in the first quadrant. The line x=7 is the axis of rotation, shown as a vertical red line.
To find the volume, we will create infinitesimally thin cylindrical shells with radius x-7 and height dx. As they rotate around the line x=7, they generate the solid.
The volume of an infinitesimally thin cylindrical shell is given by the formula V = 2πrhdx, where r is the radius, h is the height, and dx is the thickness.
In this case, the radius r is x-7, and the height h is the difference between the two curves: h = (0.5x^2) - x = 0.5x^2 - x.
Thus, the volume of each shell is dV = 2π(x-7)(0.5x^2-x)dx.
Now, we integrate this expression to find the total volume. Since the region is bounded by x=0 and x=7, the integral becomes:
V = ∫[0,7] 2π(x-7)(0.5x^2-x)dx
To evaluate this integral, expand the expression and simplify:
V = 2π ∫[0,7] [(0.5x^3 - x^2) - (7x - 7x^2)] dx
V = 2π ∫[0,7] (0.5x^3 - x^2 - 7x + 7x^2) dx
V = 2π ∫[0,7] (0.5x^3 + 6x^2 - 7x) dx
Now, integrate each term separately:
V = 2π [0.5 * (1/4)x^4 + 6 * (1/3)x^3 - (7/2)x^2] | [0,7]
V = 2π [0.125x^4 + 2x^3 - 3.5x^2] | [0,7]
Evaluate the expression at the upper and lower bounds:
V = 2π [(0.125 * (7)^4) + 2 * (7)^3 - 3.5 * (7)^2] - [(0.125 * (0)^4) + 2 * (0)^3 - 3.5 * (0)^2]
V = 2π [0.125 * 2401 + 2 * 343 - 3.5 * 49]
V = 2π [300.125 + 686 - 171.5]
V = 2π [814.625 - 171.5]
V = 2π * 643.125
V ≈ 4035.91 cubic units
Therefore, the volume of the solid obtained by rotating the region bounded by y=0.5x^2 and y=x about the line x=7 is approximately 4035.91 cubic units.