Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Question

Answered step-by-step

Find the volume of the described solid $ S $.A cap of a sphere with radius $ r $ and height $ h $.

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Chris Trentman

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

02:43

Wen Zheng

01:59

Amrita Bhasin

Calculus 2 / BC

Chapter 6

Applications of Integration

Section 2

Volumes

Missouri State University

Campbell University

Oregon State University

University of Michigan - Ann Arbor

Lectures

01:36

Find the volume V of the d…

01:13

Find the volume of a spher…

07:26

$47-59$ Find the volume of…

04:52

03:46

0:00

Find the volume of the des…

So were given a solid S. And we're asked to find the volume of this solid S. Is a cap of a sphere bon Jovi. Yeah with the radius R. Wilson and a height. Uh huh. Each there's some like and deep in yes. Now in order to um find the volume of this cap, we'll draw a circle however we'll draw it in a certain way. So set up our X. And Y axes and then we'll have a circle like this. The distance from the origin to this point on the right is going to be the height of the cap H. And so God works. The radius of the circle is going to be Yeah. Yeah. Words. Yeah. A little r We got the pork scores, murdering sucks. Yeah. All right. No I didn't even finish. But you said you're really now since the whole radius is our follows at the center of this circle is at the point let's see this is our, sorry negative are plus H. Zero. You know c. three. And therefore the equation for a circle is X. That's kind of what I wanted minus H minus R squared plus Y squared equals the radius R squared those better what? Look you can't say hard ours. Here is your anger. Anger reached a hard art. Then they got James, Earl jones to play Darth vader because he's black blue. I am now imagine rotating this across the X axis. But we end up with is our cap and we end up with disks use discs have a radius which is why equals the positive square root of r squared minus x minus h minus r squared. It was it was and shylock. So great they would play console. This is one way to approach it. I never noticed, let's say. And so the volume V by the disk method is given by the pi times the integral from x equals 02 H. Of our radius which is the square root of r squared minus x minus h minus R squared all squared dx one. So simplifying this is pi times the integral from zero to H. Of r squared minus x minus h minus or squared dx Mhm. If you take anti derivatives you should get pi times this is r squared x minus and then one third times x minus H minus R cubed From x equals zero two H. Plugging in. We get pi times R squared h -1 3rd of yeah are understood- and this is 0 -1/3 of. Sorry there should be r cubed. My mistake I as a third times negative H minus R cubed which is plus one third times h minus r cubed rider bikes at each other and then try to joe get him through the spokes data, lock the bike up, you fucking fall off. That's what and so simplifying. This is pi times R squared h minus one third R cubed. And this is actually plus 1/3 times ar- H cubed. Yeah you could leave in this form or simplify further so mhm slip behind in the bars and then got stuck. If you multiply everything out this will eventually simplify too. One third times kids are spear liss. We factor that one third pi times three R squared H minus R cubed plus ar minus H cubed. And then we get one third pie and after several steps three R h squared minus H cubed which is equal to one third pie H squared Times three AR- H dump every girl that gets her period and disrespect going to come. You gonna try in the period with me? It's

View More Answers From This Book

Find Another Textbook

00:36

QuestionWhat is the slope the line on the geoboard shown?Provide Wou…

01:53

Question Find the slope of the line shown:Provide your answer below:…

01:02

Question 25 ptsDuring spring clearance sale Sears advertises 1/4 off…

03:15

Use base ten blocks to provide three different base block representations of…

01:06

Form a polynomial whose real zeros and degree are given. Zeros: -2,0,5; degr…

00:34

AABC Is right triangle:Use the given Information to find the length of t…

02:45

Imagine cutting circle into IAHY eqqually sivxl slices aud rearranging them:…

02:12

2. Find the values of X and y The diagram 15 not t0 scale: (1 point);

01:55

Theorem 46. If two angles of a triangle are congruent, then the opposite sid…

01:22

Compute gxxyz for g(x,y,2) = xyz. (Use symbolic notation and fractions where…