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Find the volume generated by revolving the region bounded by

y2 = x, x = 4y
about the line y=5

User Zhambul
by
8.1k points

1 Answer

3 votes

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

User Qalis
by
8.0k points
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