Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.
y = 8x − x^2, y = x
The curves intersect at (0,0) and (7,7)
So, using shells of thickness dx,
v = ∫[0,7] 2πrh dx
where r=x and h=(8x-x^2)-x = 7x-x^2
v = ∫[0,7] 2πx(7x-x^2) dx = 2401π/6
To use discs of thickness dy, you need to change the boundary at (7,7) so it gets a bit more complicated.
v1 = ∫[0,7] π(R^2-r^2) dy
where R = y and r = 4-√(16-y)
v1 = ∫[0,7] π(y^2-(4-√(16-y))^2) dy = 673π/6
v2 = ∫[7,16] π(R^2-r^2) dy
where R = 4+√(16-y) and r = 4-√(16-y)
v2 = ∫7,16] π*((4+√(16-y))^2-(4-√(16-y))^2) dy = 288π
so v = v1+v2 = 673π/6 + 1728π/6 = 2401π/6
To find the volume generated by rotating the region bounded by the given curves about the y-axis using the method of cylindrical shells, we need to follow these steps:
Step 1: Determine the limits of integration.
We need to find the y-values where the two curves intersect. Setting the equations equal to each other, we get:
8x - x^2 = x
Rearranging, we have:
x^2 - 7x = 0
Factoring out x, we get:
x(x - 7) = 0
So, x = 0 or x = 7. These are the values where the curves intersect.
Thus, the limits of integration for y are from 0 to 7.
Step 2: Determine the height of each cylindrical shell.
The height of each cylindrical shell represents the difference in y-values at a given x.
The equation of the curve y = 8x - x^2 is on top, and the equation y = x is on the bottom. So, the height is given by:
height = (8x - x^2) - x
= 8x - x^2 - x
= 7x - x^2
Step 3: Determine the radius of each cylindrical shell.
The radius of each cylindrical shell is the x-value at a given y.
To express the x-value in terms of y, we rearrange the equation y = x to solve for x:
x = y
So, the radius is equal to the y-value.
Step 4: Determine the differential width of each cylindrical shell.
The differential width represents the width of each cylindrical shell.
For this problem, the differential width is an infinitesimally small change in y, which can be represented as dy.
Step 5: Set up and evaluate the integral.
The volume of a cylindrical shell is given by the formula:
dV = 2πrh dy
So, the volume generated by rotating the region about the y-axis is:
V = ∫(from 0 to 7) 2πy(7y - y^2) dy
Evaluating this integral will give us the volume of the region generated by rotating the curves about the y-axis.
I hope this helps! Let me know if you have any other questions.