Answer:
the volume generated by revolving the region bounded by
y^2 = x , x = 4y, and the line y = 5 is 96π cubic units.
Explanation:
To find the volume generated by revolving the region bounded by the curves y^2 = x and x = 4y about the line y = 5, you can use the method of cylindrical shells.
First, let's find the points of intersection of the two curves. Set y^2 = x equal to x = 4y:
y^2 = 4y
Now, rearrange to solve for y:
y^2 - 4y = 0
Factor the left side:
y ( y - 4 ) = 0
This equation has two solutions: y = 0 and y = 4
So, the region of interest is bounded by the curves y^2 = x , x = 4y and the lines y = 0 and y = 4.
Now, let's set up the integral for the volume using cylindrical shells. The volume V is given by:
V = 2π ∫ (upper b, lower a) R ( x ) h ( x ) dx
where lower a and upper b are the bounds of integration, R ( x ) is the distance from the axis of rotation (in the case, y = 5) to the outer curve, and h ( x ) is the height of the shell.
The outer curve is x = 4 y, and the inner curve is y^2 = x, so R ( x ) = 4y - 5.
THe height of the shell is the difference between the upper and lower y values which is 4 - 0 = 4.
Now we can set up the integral:
V = 2π ∫ [upper 4, lower 0] (4 y - 5) ( 4 ) d y
Now, integrate with respect to (y:
V = 2π (48)
V = 96π cubic units
So, y = 5 is 96π cubic units.
Thank you,
Jarrett Lane