Determine the volume of the solid abtained when the region bounded by y=sqrt x and the line Y=2 and x=0 is rotated:
(i) about the x-axis
(ii) about the line x=4.
(i) Each element of the body of revolution with thickness dx, has area
pi*(sqrt x)^2 dx = pi x dx.
Integrate that from x=0 to x = 4. (The sqrt x curve ends at x=4, y=2.) I get a volume of 8 pi.
(ii) The x=4 vertical line is at the end of the segment of the curve from x=0 to x=4. Perform an integration aloing the y axis. Each slab of thickess dy has volume
pi (4-x)^2 dy. Substitute y^2 for x and the integral becomes
(INTEGRAL sign) pi (4- y^2)^2 dy from y=0 to y=2.
To determine the volume of the solid obtained when the region bounded by y = sqrt(x), the line y = 2, and x = 0 is rotated about the x-axis:
(i) We can use the method of cylindrical shells to find the volume. Each cylindrical shell, with thickness dx, has an infinitesimal length given by the curve y = sqrt(x), and its radius is equal to y = sqrt(x). The height of the cylindrical shell is given by the difference between the line y = 2 and the curve y = sqrt(x), which is 2 - sqrt(x).
The volume of each cylindrical shell is given by V = 2π · (2 - sqrt(x)) · sqrt(x) · dx.
To find the total volume, integrate this expression from x = 0 to x = 4:
V = ∫(0 to 4) 2π · (2 - sqrt(x)) · sqrt(x) · dx.
Evaluating this integral will give us the volume of the solid, which is 8π.
(ii) To rotate the region about the line x = 4, we can use the method of slicing. Each slice of the solid, with thickness dy, is perpendicular to the y-axis. The distance of each slice from the line of rotation (x = 4) is given by the difference between 4 and the x-value of the curve y = sqrt(x), which is (4 - x).
The volume of each slice is given by V = π · (4 - x)^2 · dy.
To find the total volume, integrate this expression from y = 0 to y = 2, substituting y^2 for x in the equation:
V = ∫(0 to 2) π · (4 - y^2)^2 · dy.
Evaluating this integral will give us the volume of the solid when rotated about the line x = 4.