A integrating volume question giving me trouble

[MoeMousta]

Quick summary to make up for my terrible handwriting. So the question asks me to find the height (h) that will give me a quarter of the volume when I integrate. I know I have to get the volume into some type of form of integral of pi * r^2 but I am unsure of how to proceed. theta is not given so using a= r*sin(theta) doesn't help me much. I also tried setting r= h+ r*cos(theta) but that didn't help me much. Stumped on how to start this one so any help is appreciated thank you!

 

[MoeMousta]

Here is the exact question's wording

 

[Harry_the_cat]

Have you learnt about volumes of revolution? You need to first set up the equation of an approproate circle.

 

[blamocur]

Here is the exact question's wordingView attachment 32985

Are you familiar with spherical coordinates and using them in integrals?

 

[MoeMousta]

So I am learning about volume of revolutions right now. Here is the work I have done so far. What I am really stuck on is how to get the equation for the radius of what I will be rotating. Thank you for the help!

 

[Harry_the_cat]

Consider a circle with centre at the origin and radius 3. What is it's equation? Draw the circle on a set of axes.

If you rotate that around the x-axis you will get a sphere. Yes? What will it's volume be? ( You can do this the easy way seeing you already know the formula for the volume of a sphere OR the hard way by finding the volume of revolution of the whole circle - I'd suggest the easy way). In fact your working shows you have already done that.

Now, you want a quarter of that volume in the spherical cap. Again you have already worked out what that volume is to be.

(For the moment, forget the variables you have been given in the diagram you were given.)

Draw a vertical line through your circle at say x=b, which will cut of a cap of that volume. Obviously it will be just approx at this stage.

Can you think of what to do next?

 

[MoeMousta]

So this is what I tried to do based off what you said. Pretty sure this isn't the right path because the final answer doesnt give a quarter of the volume but going to try other approaches and add them here.

 

[blamocur]

I believe you got the sign wrong.

 

[MoeMousta]

Still no luck even when I swapped the signs. I tried setting the integral from some random point (b) to the radius 3 and then using 3-b=h but that isn't giving me a number that gives me the expected volume

 

[Harry_the_cat]

Yes that is correct. h=1.96 approx.

How are you testing your answer? Why do you say it is incorrect?
If you rotate the part of the circle from x=1.0419 to 3 you do get 9pi.
(You are not rotating it from x=1.9581 to 3 are you?)

Another way to check your answer is correct is to use the formula for the volume of a spherical cap:
\(\displaystyle V = \frac{1}{3}\pi h^2(3r - h)\) where h and r are defined as in your given diagram.