Question 3: Explain how to find the limits of integration in spherical coordinates in the order d? d? d?. Notice that just like how we have an extra factor of r when integrating in polar or cylindrical coordinates, we have an extra factor of ? 2 sin(?). Note: This font uses the cursive phi ?, which is written as ? in the textbook. These are the same letter. This is for similar, though more complicated, reasons that we had with polar coordinates.

Hint 1: Read the subsection "How to Integrate in Spherical Coordinates" (p. 947 - 949).

15.7 Triple Integrals in Cylindrical and Spherical Coordinates 947 attention to integrating over domains that are solids of revolution about the z-axis (or por- tions thereof) and for which the limits for # and are constant. As with cylindrical coordi- nates, we restrict o in the form a = # s Band 0 = B - q = 2x. How to Integrate in Spherical Coordinates To evaluate over a region / in space in spherical coordinates, integrating first with respect to p, then with respect to $, and finally with respect to 8, take the following steps. 1. Sketch. Sketch the region D along with its projection R on the xy-plane. Label the surfaces that bound D. P = (d. 8) D Sp - gi(d, 8)2. Find the p-limits of integration. Draw a ray M from the origin through D. making an angle & with the positive z-axis. Also draw the projection of M on the xy-plane (call the projection L). The ray L makes an angle d with the positive x-axis. As p increases, M enters D at p = gi(6, 8) and leaves at p = gy(6, 8). These are the p-limits of integra- tion shown in the above figure. 3. Find the o-limits of integration. For any given 8, the angle that M makes with the Z-axis runs from