Volume of a Solid of Revolution?Find the volume if the region enclosing y = x...

Dean Summers

Dean Summers

Answered

2022-07-17

Volume of a Solid of Revolution?
Find the volume if the region enclosing y = x 3 , x = 0 ,, and y = 8 is rotated about the given line. So the axis of rotation that was given was the y axis, but I'm a bit confused on what the given bounds of y = 8 and x = 0 have to do with it, but I pretty much ignored them and put the given equation in terms of y and used the integral 0 7 ( 2 2 ( y 3 ) 2 ) to solve, is this at all right? Any help would be appreciated Also looking for some guidance on how I'd go about finding the volume if the axis of rotation was something like x = 6.

Answer & Explanation

Rihanna Robles

Rihanna Robles

Expert

2022-07-18Added 18 answers

Step 1
Note that y = x 3 , x = 0 and y = 8 only specify the area to be rotated. The volume is then defined by rotating this area around a given axis.
Step 2
For instance, if the axis of rotation is x = 6, the volume can be integrated with the disk method 0 8 π [ 6 2 ( 6 y 3 ) 2 ] d y
Glenn Hopkins

Glenn Hopkins

Expert

2022-07-19Added 4 answers

Step 1
Now, we can go about finding the volume. Since we are revolving around the y-axis, we can use the following formula: V = a b A ( y ) d y
And, we know that A ( y ) = π ( ( outer radius ) 2 ( inner radius ) 2 ), where the radius is just the distance from the curve to the axis of rotation.
Since the axis of rotation is the y-axis (equivalently, x = 0), we know that are distances must be x-distances, so we must solve our y-functions in terms of x.
Our outer radius is just the distance from the curve y = x 3 to the y-axis, we which is given by y 1 / 3 0 = y 1 / 3 .
Step 2
Our inner radius is the distance from the y-axis to itself, which is 0.
Therefore, the expression for the area in terms of y is:
A ( y ) = π ( ( y 1 / 3 ) 2 ( 0 ) 2 ) = π ( y 2 / 3 )
Now, we just need to plug back into the equation for the volume, and figure out the bounds. Since this is a dy integral, we need to know the y-bound. From the region, we can see that the integral should go from 0 to 8, since the region is bounded by y = 0 and y = 8
Therefore, our integral becomes: V = 0 8 π ( y 2 / 3 ) d y
If the axis of rotation changes, you just need to change your expression for the area. For example, for an axis of rotation of x = 6, you can keep the area in terms of y, but now your inner and outer radii expressions will be different.

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