Work on the following:
30. Evaluate the following integrals: a. sin (ma3) dx dy b. [, ye s' dA where R= (ry) :0sysx) 2 V2y-y- cy C. dax dy 0 dy dx d. S. (x2+ 12)2+128. Three hemispheres with radiuses 1, a and y, where 1 2 x 2 y 2 0, are stacked on top of each other as shown in the figure. Find the largest possible value of the total height h. 2 2 X- X h 1/1-x2 11 31. Evaluate the double integral dA where R is the region shown in the figure. (2 + 1/2) 2 - 1- R - 114. Once upon a time there was an ry-plane. The temperature at each point of this ry-plane changed as time passed. Bugs roamed this ry-plane. Each bug had a device that measured the temperature in real time, and showed the rate of change of temperature with respect to time on its screen. One day four of these bugs met at a point Po. At the moment they met: . The first bug was moving with velocity v1 = 2i + j m/s and its device was showing 1.C/s. . The second bug was moving with velocity v2 = i -5j m/s and its device was showing -1 -C/s. . The third bug was moving with velocity v3 = i+ j m/s and its device was showing 2-C/s. . The fourth bug was sitting still. What was the device of the fourth bug showing?32. Evaluate the following integrals. pay cos(my /2) dA where R is the region shown in the figure. a. xlycos(myS/2) dA where R2 is the region shown in the figure. b. 1 = 1 R2 RI19. Consider the following conditions for a diiferentiahle function f[or,y] : i 1121]] = 3 E An equation for the tangent line to the level curve ay] = 3 in the :ryplane at the point [2,1] is 3355:: 1 Let 9 he the tangent plane to the graph of 2: = nely] at the point [2,1,3]. In each ofthe parts [ae] below a a\" condition is given. Determine whether a there is no function satisfying the conditions IB'! - there are functions satisfying the conditions Ill-E! but they do not all have the same tangent plane '93, or a there are functions satisfying the conditions ill and all of these functions have the same tangent plane 33. [In this case nd an equation of .9 too] a. f[3,2]=11 s. a filmy1 s c. o Et+lfjt= =6 1 d. 9 The line with parametric equations I = 411+? , y = 2f+1, z = t+3,[oo r: f