Can you help me with problems 28, 32, 36 from the last page? Thanks so much!

. AT&T 10:23 PM 01 9% SECTION 11.3 Polar Coordinates 637 penises 31-38. use Eq. (4) to compute the surface area of the given 36. The surface generated by revolving the curve c(!) = (f. sint) about ux the x-axis for 0 $ I S . Him: After a substitution. use #84 in the table in the mee generated by revolving c(f) = (f, my) about the x-axis for the front or back of the text for the integral of v1 + w-. 37. The surface generated by revolving one arch of the cycloid c() = (f - sint. 1 - cos/) about the x-axis A sphere of radius R 38. The surface generated by revolving the astroid c() = (cost. sin') The surface generated by revolving the curve c(f) = (12.() about the about the x-axis for 0 Is 39. (CAS) Use Simpson's Rule and N = 30 to approximate the surface me surface generated by revolving the curve c(1) = (f. e') about the area of the surface generated by revolving c(!) = (r. e=).0 0 and set k = 2vab a - b' Show that the trochoid 8 = Gme - 9.8 m/s. where Re = 6378 km Use Exercise 43(b) to show that a satellite orbiting at the earth's surface x = al - bsint. y = a - bcost. OSIST would have period Te = 2n VR./g * 84.5 minutes. Then estimate the dis- tance Rm from the moon to the center of the earth. Assume that the period s length 2(a - b)G (. k), with G(@. k) as in Exercise 30. of the moon (sidereal month) is Tm * 27.43 days. da coordinates are appropriate when save from the origin or angle plays a 11.3 Polar Coordinates For example, the gravitational force ended on a planet by the sun depends The rectangular coordinates that we have utilized up to now provide a useful way to rep- on the distance r from the sun and is resent points in the plane. However, there are a variety of situations where a different zanently described in polar coordinate system is more natural. In polar coordinates, we label a point P by coor- ordinates. dinates (r, 0), where r is the distance to the origin O and 0 is the angle between OP and the positive x-axis (Figure 1). By convention, an angle is positive if the correspond- ing rotation is counterclockwise. We call r the radial coordinate and o the angular P = [(x. y) (rectangular) coordinate. (r. 0) (polar) The point P in Figure 2 has polar coordinates (r, 0) = (4, " ). It is located at dis- tance r = 4 from the origin (so it lies on the circle of radius 4), and it lies on the ray of angle 0 = . Notice that it can also be described by (r, 0) = (4, =). Unlike Carte- ly = r sin e sian coordinates, polar coordinates are not unique, as we will discuss in more detail shortly. Figure 3 shows the two families of grid lines in polar coordinates: I* r Co5 8 Circle centered at 0 + r = constant FIGURE 1 Ray starting at O + 0 = constant 1/8. AT&T 10:23 PM 01 9% 638 CHAPTER 11 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS Every point in the plane other than the origin lies at the intersection of the two grid to and these two grid lines determine its polar coordinates. For example, point Q in Fires lies on the circle r = 3 and the ray 0 = . so 0 = (3. ) in polar coordinates. Figure I shows that polar and rectangular coordinates are related by the equa, x = rcose and y = r sine. On the other hand. "? = x2 + yz by the distance for and tano = y/x if x # 0. This yields the conversion formulas: Polar to Rectangular Rectangular to Polar x = r cose r = Vx2 + y2 y = r sine tan 0 = = (x #0) FIGURE 2 Note, we do not write 0 = tan" since that relationship holds only for - ; 0, ad-coordinate of P = (x. y) is . The angular coordinate is not unique because (r, 0) and (r, 0 + 2:Ta) labd h tan -12 same point for any integer n. For instance, point P in Figure 5 has radial cou if x > 0 dinate r = 2, but its angular coordinate can be any one of J. .... or- 0 = tan - +x ifx ) represent the same point. . We may specify unique polar coordinates for points other than the origin s placing restrictions on r and e. We commonly choose r > 0 and 0 s but other choices are sometimes made. 2/8. AT&T 10:23 PM 01 9% SECTION 11.3 Polar Coordinates 639 P = (0. 2) (rectangular) (-r. 0) or (r. 8 + 2) FIGURE 5 The angular coordinate of P = (0,2) is FIGURE 6 Relation between (r.) and or any angle # + 2xn, where n is an integer. ( -r.0 ). Pet-1, 1) When determining the angular coordinate of a point P = (x, y), remember that there are two angles between 0 and 27 satisfying tan 0 = y/x. One of these angles is paired with r = vx2 + y? to provide polar coordinates for P. the other is paired with -r. EXAMPLE 3 Choosing @ Correctly Find two polar representations of P = (-1, 1). (1.-1) one with r > 0 and one with r 0. then tan-1 _ if x > 0 0 = tan- - +m ifx