1 Calculate the iterated integral 1 1 2 Find the volume of t

1 Calculate the iterated integral 1 1 2. Find the volume of the solid which is above the region in the xy plane bounded by 0, x 3. Use polar coordinates to find the volume of the solid above the cone z x2 y2 and below the sphere x2 y z 4. Evaluate the triple integral below where E lies under the plane z 1 x y and above the region in the xy-plane bounded y Vx, y 0, and x 1. 6xy dv 5.Use cylindrical coordinates to evaluate x2 dV where E is the solid that lies within the 4x2 4y 0, and below the cone z cylinder y 1, above the plane z 6.Use spherical coordinates to find the volume of the solid that lies within the sphere x2 y 2 z 25, above the xy plane and below the cone z x2 y

Solution

Using cylindrical coordinates,
z² = 4x² + 4y² ==> z = ± 2r. (In this case, we ignore the negative sign since z 0.)

The projection of E into the xy-plane is the unit disk x² + y² 1
==> r 1 with in [0, 2].

Thus, E x^2 dV
= ( = 0 to 2) (r = 0 to 1) (z = 0 to 2r) (r cos )² (r dz dr d).

Evaluating this yields
( = 0 to 2) (r = 0 to 1) (z = 0 to 2r) (r³ cos² ) dz dr d
= ( = 0 to 2) (r = 0 to 1) (r³ cos² ) (2r - 0) dr d
= [( = 0 to 2) 2 cos² d] * [(r = 0 to 1) r dr]
= [( = 0 to 2) (1 + cos(2)) d] * [(r = 0 to 1) r dr]
= [( + sin(2)/2) {for = 0 to 2}] * [r/5 {for r = 0 to 1}]
= 2/5.

I hope this helps!