Desperate college student needs help in double integral

[marik]

here is the problem I couldn't solve, anyone got any idea please help me.
thank you very much in advance

1) use double integrals to derive the given formula for the volume of a right circular cone of radius R and height H. the volume of a cone is given by the formula

(pi*R^2*H)/3
I tried to use polar coordinates, but what is troubling me is that I couldn't get H into the formula.


2) use double integrals to derive the given formula for the volume of a cap of a sphere of radius R and height H where 0<H<R. (the cap of a sphere is the portion of the sphere bounded below by the plane z=R-H and bounded above by the plane z=R). the volume of a cap of a sphere with radius R is given by the formula

( pi*H^2*(3R-H))/3
same problem, I couldn't get H into the formula from double integration


3) use double integrals to derive the given formula for the surface are of a cap of a sphere of radius R and height H where 0<H<R. (the cap of a sphere is the portion of the sphere bounded below by the plane z=R-H and bounded above by the plane z=R). the surface area of a cap of a sphere with radius R is given by the formula

2*pi*R*H
same problem, I couldn't get H into the formula from double integration


4) use a double integral to calculate the area for the region in xy-plane bounded by y=H, y=0, x=0, and the line containing the point (a,0), and (b,H) where a,b,H>0 and b<a.


5) use double integral to calculate the area for the sector in the polar plane bounded by the ray thetha=0 and thetha=R>0 and the circle x^2+y^2=R^2 in the first quadrant.


any help on any problem is deeply appreciated

 

[eok20]

For #1, using polar coordinates is a good idea. You should be able to see that the height , i.e. z, of a part of the cone is a function of r. This function will have H in it. If you're uncertain how to find it, you should be able to see that its is linear and what are its values when r=0 or r=R.