1. ## Integration #2

a) Find the area of the circle formed when a sphere is cut by a plane at a distance y from the centre, where y > r.
I worked this out to be $\displaystyle \pi(r^2 - y^2)$.
b) By integration, prove that the volume of a 'cap' of height $\displaystyle \frac{1}{4}r$ from the top of the sphere is $\displaystyle \frac{{11\pi}r^3}{192}}$.
I'm not really sure what to do here. I can find the area of the circular base of the 'cap', but I don't know what to do with it.

Thanks.

2. ## Re: Integration #2

What you might want to do is take the area of a vertical cross-section of the cap and rotate that through $2 \pi$ radians -- google up on "solid of revolution" for background. When I was at school (hrumphity years ago) it was taught in advanced applied mathematics at the grade one is at when one is aged 17-18.

3. ## Re: Integration #2

That would work, but how could I find the value of the base of that cross-section?

4. ## Re: Integration #2

Originally Posted by Fratricide
a) Find the area of the circle formed when a sphere is cut by a plane at a distance y from the centre, where y > r.
I worked this out to be $\displaystyle \pi(r^2 - y^2)$.
I assume you mean that r is the radius of sphere and y< r, not y> r.

b) By integration, prove that the volume of a 'cap' of height $\displaystyle \frac{1}{4}r$ from the top of the sphere is $\displaystyle \frac{{11\pi}r^3}{192}}$.
I'm not really sure what to do here. I can find the area of the circular base of the 'cap', but I don't know what to do with it.

Thanks.
Think of the "cap" as a series of disks at distance "y" from the center of the sphere, y going from (1/4)r to r, with thickness "dy". What is the area of a circle at distance y from the center of the sphere? What is the volume of such a disk?

5. ## Re: Integration #2

Is the area simply what I had in part (a)? And the volume the area multiplied by the thickness, dy? Are you hinting at something like $\displaystyle \pi{\int_{{\frac{1}{4}r}}^{r}(r^2 - y^2)^2 dy}$ perhaps?
The text passage contains a mathematical problem with multiple parts and a discussion around solving it. The problem involves calculating the area of a circle formed by a plane cutting a sphere and proving the volume of a spherical cap through integration.
multiple_questions_with_answers

1. the first part of the problem asks for the area of the circle formed when a sphere is cut by a plane at a distance \( y \) from the center, where \( y < r \). the answer provided is \( \pi(r^2 - y^2) \).

2. the second part of the problem asks to prove that the volume of a 'cap' of height \( \frac{1}{4}r \) from the top of the sphere is \( \frac{11\pi r^3}{192} \). the discussion includes hints and suggestions for solving this part, but a complete answer is not provided in the text.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
1. ## Integration #2

a) Find the area of the circle formed when a sphere is cut by a plane at a distance y from the centre, where y > r.
I worked this out to be $\displaystyle \pi(r^2 - y^2)$.
b) By integration, prove that the volume of a 'cap' of height $\displaystyle \frac{1}{4}r$ from the top of the sphere is $\displaystyle \frac{{11\pi}r^3}{192}}$.
I'm not really sure what to do here. I can find the area of the circular base of the 'cap', but I don't know what to do with it.

Thanks.

2. ## Re: Integration #2

What you might want to do is take the area of a vertical cross-section of the cap and rotate that through $2 \pi$ radians -- google up on "solid of revolution" for background. When I was at school (hrumphity years ago) it was taught in advanced applied mathematics at the grade one is at when one is aged 17-18.

3. ## Re: Integration #2

That would work, but how could I find the value of the base of that cross-section?

4. ## Re: Integration #2

Originally Posted by Fratricide
a) Find the area of the circle formed when a sphere is cut by a plane at a distance y from the centre, where y > r.
I worked this out to be $\displaystyle \pi(r^2 - y^2)$.
I assume you mean that r is the radius of sphere and y< r, not y> r.

b) By integration, prove that the volume of a 'cap' of height $\displaystyle \frac{1}{4}r$ from the top of the sphere is $\displaystyle \frac{{11\pi}r^3}{192}}$.
I'm not really sure what to do here. I can find the area of the circular base of the 'cap', but I don't know what to do with it.

Thanks.
Think of the "cap" as a series of disks at distance "y" from the center of the sphere, y going from (1/4)r to r, with thickness "dy". What is the area of a circle at distance y from the center of the sphere? What is the volume of such a disk?

5. ## Re: Integration #2

Is the area simply what I had in part (a)? And the volume the area multiplied by the thickness, dy? Are you hinting at something like $\displaystyle \pi{\int_{{\frac{1}{4}r}}^{r}(r^2 - y^2)^2 dy}$ perhaps?
user
The text passage contains a mathematical problem with multiple parts and a discussion around solving it. The problem involves calculating the area of a circle formed by a plane cutting a sphere and proving the volume of a spherical cap through integration.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS

1. The first part of the problem asks for the area of the circle formed when a sphere is cut by a plane at a distance \( y \) from the center, where \( y < r \). The answer provided is \( \pi(r^2 - y^2) \).

2. The second part of the problem asks to prove that the volume of a 'cap' of height \( \frac{1}{4}r \) from the top of the sphere is \( \frac{11\pi r^3}{192} \). The discussion includes hints and suggestions for solving this part, but a complete answer is not provided in the text.