(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

We have to trace a pumpkin on graph paper and then find it volumne when rotated around the y-axis. Upon doing so we have 2 pieces we can do. One is a semi circle and the other just a rectangle. Refer to this image:

Where we can see the diameter of the circle is 13.5 and the full width of the shape is 9.

2. Relevant equations

(x-h)^2 + (y-k)^2 = r^2

v = integral from a to b pi r^2 times thickness (dx or dy)

3. The attempt at a solution

Know the diameter is 13.5, we know the radium is (13.5 / 2) or 6.75. Using that we can pinpoint the origin of the semi-circle to (2.25,6.75), where 2.25 is the width of the whole shape (9) minus the radius (6.75).

We use the formula of a circle with center h,k:

(x-h)^2 + (y-k)^2 = r^2

Thus:

(x-2.25)^2 + (y-6.75)^2 = (6.75)^2.

We know to be with respect to y there we solve for x:

x = sqrt( (6.75)^2 - (y-6.75)^2) + 2.25

Now we get into rotating around the y-axis. The semi-circle is shifted 2.25 from the y-axis so you add 2.25 to the equation, giving you a radius of sqrt( (6.75)^2 - (y-6.75)^2) + 4.5 thus:

v= pi * integral from 0 to 13.5 (sqrt( (6.75)^2 - (y-6.75)^2) + 4.5)^2 dy

which is about 1327.56 pi.

Finally just add to that the retangle rotated around y-axis which is radium 2.25:

v = pi * integral from 0 to 13.5 (2.25)^2 dy

which is about 68.3438 pi.

Answer being about 1395.9 pi.

I know this is wrong because if I just do a large rectangle of width 9 and rotate it around I get

v = pi * intefral from 0 to 13.5 (9)^2 dy

or 1093.5 pi. This should be larger that my previous answer, and it's not.

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# Homework Help: Solid volume rotation around y-axis

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