volume and calculus for upcoiming exam!

Hi,Would really appreciate help with text in red.thanks.

one of the question asks for region of integration and evaluation of e^(2x+y)dydx where inner integral goes from x to 3x;outer integral froom o to 4.

but I have issues with this part:-how should I solve?

the solid sphere x^2+y^2+z^2 is less than or equal R^2 has a hole bored through it by the solid vertical cylinder x^2+y^2 less than or equal to Q^2 where

0<Q<R.

calculate volume of part of the sphere that is removed.

Re: volume and calculus for upcoiming exam!

Hey n22.

Hint: What is the volume of a cylinder with radius Q and height Q?

Re: volume and calculus for upcoiming exam!

Hi Chiro,

do I just use basic volume formula?

so..Volume =4/3piR^3-piQ^2h

remember how I said that there was a part one for this question ?? well.. am I supposed to use that(the bit about integration) ?

thanks.

Re: volume and calculus for upcoiming exam!

You need to solve for Q given some R but once you have that then yes use the formula.

Re: volume and calculus for upcoiming exam!

You can do this in three parts- a cylinder and two "caps".

The main part is a cylinder of radius r and height given by using the Pythagorean theorem. Drawing a line from the center of the sphere to a point where the cylinder reaches the surface of the sphere, a line from the center of the sphere, perpendicular to the cylinder, and the line connecting those gives a right triangle in which the length of the hypotenuse is the radius of the sphere, one leg is the radius of the hole, and the other leg is half the length of the cylinder.

The two "caps" are the portion of the sphere above and below the cylinder. Of course, their volumes are the same so you can calculate one and multiply by two. The bottom of the cap is the top of the cylinder so you can integrate the volume of the sphere from h/2 (h being the length of the cylinder above) to the radius of the sphere.