Ask your own question, for FREE!
Mathematics 66 Online
OpenStudy (tim):

a hole of radius ""b" is drilled inside a sphere of radius "a" find the remaing volume

OpenStudy (anonymous):

So let's do it: 1. Off course we can use double integral, but let's use single integral with defined limits. 2. Integral (f(x), a, b) means definite integral of fuction f in interval [a, b] a) Let's make uppper side of circle which during rottion in OX give us a sphere in functional view. we can define it like this: f(x)=+sqrt(R^2 - x^2) Integral(PI*((f(x)*)^2)dx,-R, +R) give us a volume of a sphere (You can imagine this of make some real mathematics calcultion to prove it. It's integral summ of simple cylinders) b) We want to substract cylinder from this spere. Let this radius be r. If you imagine the picture. You'll see that a point from which starts "cap" can be calculated like this: f(x) = r <=> r^2 = R^2-(x')^2 <=> (x') = +-sqrt(R^2-r^2); the cylinder can be defined like this: we rotate g(x)=r where x is [-sqrt(R^2-r^2), +sqrt(R^2-r^2)] c) So total volume is Integral(PI*((f(x)*)^2)dx,-x', +x') - Integral(PI*((g(x)*)^2)dx,-x', +x') = Volume. Let's clculate this. Pi* Integral( (R^2-x^2) - r^2)dx, -x', +x') = ... Notice: make variable change x=sin(t) * (x'); dx=cos(t)dt*(x') and all be ok. ----------------------------------------------------------------------------------- If you need help visualizing the cylandrical shells method, take a look at http://mathdemos.gcsu.edu/mathdemos/shellmethod/gallery/gallery.html Just imagine that your taking the sphere volume using a bunch of cylandrical shells, but them leaving a bunch of them out for the hole in the middle. So once you have you're integrated function (4Pix^3)/4), use values R (for the radius of the circle) and r (radius of the cylander) like so: (4Pi(R)^3)/4)-( 4Pi(r)^3)/4)

Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours!

Join our real-time social learning platform and learn together with your friends!
Latest Questions
JustKOI: a poem just becauuuse...
2 hours ago 5 Replies 10 Medals
madz2345: random stuff from my sketchbook. rate them :0
28 minutes ago 34 Replies 11 Medals
Fredyy: Do you all im doing better!!!pls sub it helps alot!!! thx
41 minutes ago 16 Replies 6 Medals
Tyrun: What y'all think of these lyrics. its sad rap
2 hours ago 10 Replies 2 Medals
lovelygirl13: i made a rick and morty collage
53 minutes ago 24 Replies 9 Medals
DatBoiAYD: art
15 hours ago 14 Replies 6 Medals
Sharkattack123: Congrats on BLUE DEATHHHHHHHH
2 hours ago 17 Replies 6 Medals
Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours!

Join our real-time social learning platform and learn together with your friends!