A circle x^{2}+y^{2}=a^{2} is rotated around the y-axis to form a solid sphere of radius a.

Brenton Dixon 2022-07-23 Answered
Finding volume of a sphere
I am stuck on the following problem:
A circle x 2 + y 2 = a 2 is rotated around the y-axis to form a solid sphere of radius a. A plane perpendicular to the y-axis at y = a 2 cuts off a spherical cap from the sphere. What fraction of the total volume of the sphere is contained in the cap?
So far I have figured out the following:
Rotating the cap on the y axis we get a height h starting from y = a 2 . The interval from y = 0 to y = a 2 (the region below the cap) should be:
a h
I also know that the radius of the sliced disk, x, can be derived from the equation of the circle:
x = a 2 y 2
Since the area of a circle is A = π r 2 the area with respect to y for the circle should be:
A ( y ) = π ( a 2 y 2 )
So to find the volume, we need to integrate the function:
V = a 2 a π ( a 2 y 2 ) d y
I know where I should go, but I am not sure what to do about the constraint y = a 2 at this point. Should I integrate the terms with respect to y first and then plug in the value which is equal to y? Or should this be done before integrating?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Sandra Randall
Answered 2022-07-24 Author has 17 answers
Step 1
y = a / 2 is not a constraint; it's one of the integration bounds, and you've already written it into the integral as an integration bound correctly. Now all you have to do is evaluate the integral.
Step 2
By the way, your derivation of the bounds seems unnecessarily complicated. There's no reason to invoke y = 0, which doesn't play any special role here. The truncated sphere lies between -a and a/2 and the cap that's cut off lies between a/2 and a, so you integrate from a/2 to a; nothing to be subtracted or calculated there.

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-08-06
Volume of a Hyperboloid using triple integration/shadow method
Could someone help me find the volume of the hyperboloid
x 2 1817 + y 2 1817 z 2 10914 = 1
with the limits in the z a x i s of -130 to 43? I tried using triple integration, but I'm not sure how to convert the variable the in limit of the innermost integral into a number. I would also be open to any other methods of finding the volume.
asked 2022-08-26
Finding volume of the solid
The region bounded by the curve y = x , the x-axis and the line x = 4 is revolved about y-axis to generate a solid. Find the volume of solid.
I found 32 π / 5? But answer is wrong. Where did I wrong?
Volume   y = π 0 2 [ A ( y ) ] 2 d y = π 0 2 y 4 d y
asked 2022-08-16
Intuitive error in finding Volume of sphere using single integration.
To calculate the volume of a sphere of radius R, I considered a thin disc of radius r = R s i n θ, where θ is the angle radius vector on the circumference makes with the axis of the disc. However I considered the volume d V = π r 2 R d θ, R d θ being the infinitesimal thickness of the disc, which is erroneous. The volume of the sphere comes out to be 3 π 2 R 3 2 . The correct thickness is R s i n θ d θ which is found usually in other derivations gives the correct volume. However I do not understand what exactly is the issue in my approach and why the component of the curve length is considered.
asked 2022-07-26
What is the volume of a right right pyramid whose base is a square with a side 6m long and whose altitude is aqual to base side? wich one is A 36M OR B 72M OR C 108M OR D 216M 3 SQUARE WICH ONE IS IT
asked 2022-07-14
Gauss-divergence theorem for volume integral of a gradient field
I need to make sure that the derivation in the book I am using is mathematically correct. The problem is about finding the volume integral of the gradient field. The author directly uses the Gauss-divergence theorem to relate the volume integral of gradient of a scalar to the surface integral of the flux through the surface surrounding this volume, i.e.
C V ϕ d V = δ C V ϕ d S
asked 2022-07-26
Let Vr denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r ? 1). Let r = TVr+1 ? Vr ? 2 and Qr = Tr+1 ? Tr for r = 1, 2,
asked 2022-07-15
When do we use 2 π vs using π in finding the volume of a region?
I've seen that sometimes finding the integral volume of a rotated shape in the xy-plane is multiplied by 2 π, and other times it is only multiplied by π. Can someone please tell me the difference?

New questions