Find the volume of the solid produced by revolving the region enclosed by y=4x and y=x^{3} in the first quadrant.

Marisol Rivers

Marisol Rivers

Answered question

2022-07-14

Finding volume of solid in one quadrant - divide total volume by 4? 8? 2?
I want to find the volume of the solid produced by revolving the region enclosed by y = 4 x and y = x 3 in the first quadrant. The wording about the first quadrant confuses me but here's my work so far:
I know the volume unrestrained by quadrant is:
V = a b π ( f ( x ) 2 g ( x ) 2 ) d x
Where f ( x ) = 4 x and g ( x ) = x 3 . To find a and b, I look for the largest and smallest intersection points between the two functions:
f ( x ) = g ( x ) 4 x = x 3 0 = x 3 4 x = x ( x 2 ) ( x + 2 ) x { 2 , 0 , 2 }
Plugging all of these into the volume equation above:
V = a b π ( f ( x ) 2 g ( x ) 2 ) d x = 2 2 π ( ( 4 x ) 2 ( x 3 ) 2 ) d x = π 2 2 ( 16 x 2 x 6 ) d x = π ( 2 2 16 x 2 d x 2 2 x 6 d x ) = π ( [ 16 x 3 3 ] 2 2 [ x 7 7 ] 2 2 ) = π ( 16 ( 2 ) 3 3 16 ( 2 ) 3 3 ( 2 ) 7 7 ( 2 ) 7 7 ) = π ( 128 3 + 128 3 128 7 + 128 7 ) = π ( 256 3 256 7 ) = π ( 1024 21 ) = 1024 π 21
This is the volume for the entire function. I make an assumption that since I only want one quadrant and the function is symmetric about both the x- and y-axes, I simply divide it by four.
V whole = 1024 π 21 V one quadrant = 1024 π 21 × 1 4 = 256 π 21
I have no way of verifying my results. Can my assumption be made, or there's a differing method I should be using here?
If I'm now working in 3D space, would I instead divide it by eight? But if I'm revolving around x = 0, wouldn't the solid of revolution take four quadrants in 3D space, thus I should divide the total volume by 2?

Answer & Explanation

constanzma

constanzma

Beginner2022-07-15Added 8 answers

Step 1
The first quadrant is defined by { ( x , y ) R 2 | x , y 0 }. The instead of > is debatable, but is of no consequence for this problem.
Thus your intersection points are (0,0) and (2,2). So, the volume integral would be 0 2 .
Step 2
To answer your other question, it would be twice the volume of the correct volume, since you are integrating an even function ( f 2 g 2 ) from -2 to 2. Intuitively you are doubling the volume by calculating the volume of the identical rotated ( π radians) region between -2 and 0.

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