Find the volume of the solid enclosed by the surfaces x² + y² + z² = a , x² + y² +z² = b , (a<b) and z = (x² + y² )½ .​

An obvious substitution would be to take u = x - y and v = x + 2y, so that 0 < u < 1 (which follows from y = x   ==>   x - y = 0 and y = x - 1   ==>   x - y = 1), and 0 < v < 1.

Solve for x and y in terms of u and v :

u - v = (x - y) - (x + 2y) = -3y   ==>   y = (v - u)/3

2u + v = 2 (x - y) + (x + 2y) = 3x   ==>   x = (2u + v)/3

Compute the Jacobian and its determinant:

Now in the integral, we have

1) c 2143.6 cm

2) c 113.0 m

3) a 33.5 ft

4)   d 2 cm

5) c 20 mm

6) 180 in

7) d 120 ft

answer: sine

step-by-step explanation: