2.
Use a triple integral in spherical coordinates to find the volume of the solid bounded
by the spheres x² + y² + z² = 1 and x² + y²+ z² = 4 and the planes x - y = 0, √√3x - y = 0
and z = 0. Give exact answer only, no decimals. Be sure to sketch the base of the solid in the
xy-plane. Also, be sure to write the domain of integration in set notation for the integral in
spherical coordinates only. Note: Need to show how you find the polar angles. For
this, two of the given planes project lines onto the xy-plane. Write the equations of
those planes in the slope-intercept form and obtain the polar angles by considering
that the slope of the line is the tangent of the angle, thet is m = tane. No need to
show calculations for the azimuthal angles as they are obvious.
y บ
3
2
3 +2 +1
3
O
Φ
1
2
3 x