What are the steps to solve these difficult questions?

[riot]

past exams questions...help~!

hi i encountered a few difficult questions and i don't know how to solve them.. any help would be greatly appreciated:

1.
Find the area of the cap cut from hemisphere X^2 + Y^2 +Z^2=2 (Z=>0)
by the cylinder X^2 + Y^2=1

2.
Use method of undetermine coefficient to solve the linear system
dX/dt= X - Y + 1
dY/dt=2X + 4Y -2

 

[siddharth]

What are your thoughts/ideas on these questions? You need to show some work before you get help.

 

[Benny]

For question one do you mean the surface area? Just parameterise the surface using the standard (whichever one it is that you were taught to use) parameterisation for a sphere of radius sqrt(2). (You should know why you need to consider a parameterisation for a sphere of radius sqrt(2).)

$$\Phi \left( {\theta ,\phi } \right) = \left( {\sqrt 2 \sin \theta \cos \phi ,\sqrt 2 \sin \theta \sin \phi ,\sqrt 2 \cos \theta } \right)$$

Phi is the polar angle (the one in the xy plane). Now all you need to do is determine the limits of theta and phi and then you can calculate the surface area as you usually would.

At first glance I would say that you would need to consider an appropriate triangle to find the limits of theta. (The limits of phi are obvious, it's just 0 <= phi <= 2pi.) To do this you'll probably want to find the z-value/s which correspond to the intersection between the sphere and cylinder.

x^2 + y^2 + z^2 = 2...(1)
x^2 + y^2 = 1...(2)

(2) into (1) gives 1 + z^2 = 2 which implies z = 1 since z>=0. Now just consider the appropriate triangle to find the 'range' of the theta values.