Summer, 2012 Math22lO Midterm 3 fr/ Name Dylan Zwick 7.-zo/ Date / Instructions: Please show all of your work as partial credit will be given where appropriate, and there may be no credit given for problems where there is no work shown. All answers should be completely simplified, unless otherwise stated. 1. Evaluate the integrals. (a) f fxy+y2dydx (10 points) SI $1 1 ( x± (+i 2L_dJ 4 Answer 1(a): rr/2 (b) (15 points) 0 2yz • x dxdydz If’ sin(—) y 0 sin2z 0 Ly Jo )dy&( : yZ (oY(Z?) I 0 HU S’Ui 2-) Jo I + - Jo / 1 1 J (4 :JL 1 ¶/ Q Answer 1(b): i{: ) (-I ± (Note: This is #1 continued!) f f exckdy (c) (20 points) ord 7 cT y e z 0 L ([- -) -e) z Answer 1(c): 2. (15 points) Calculate the area of the region inside the circle outside the circle r=2 Hint: 2 cos 0= 1+cos2O r=4cos0 L1cse;, cC5 -)d y Z Ctccy9 t 1 (cy Answer 2: 2 1 L J and 3. (15 points) If R((x, y):Ox6,Oy6) and P is the partition of R into nine equal squares by the lines x = 2, x = 4, y = 2, and y = 4. Approximate Rf(- y)dA by calculating the corresponding Riemann sum If f(xk, y) A Ak , assuming that (k’ Yk) are the centers of the nine squares.Take f(x,y)=6+2x+3y A /7 .( I, j) /f (I,3) (xi)?5 43)DZ/ (i,5) (7,5) 27 H(1i+i’ii 7f?/? +2*27y i( 5 75 Answer 3: I I) - -D = >z( I \ (_ :: a) Lfl( t3 -D (13 G) -c 0 -I-, 4- t 0 a) -c o I. 4- (13 (3 L)1 til a) .(-J (13L —a) (13 LI1 c ow o--D Lfl (N c >yJ Ir.J + — IC— iJI’ H •— p I Hs H —‘ - r U, C 2--x 2 2 9—x 5. (20 points) Rewrite the integral of integration to f ( x y, z) dz dy dx changing the order , dzdxdy y y L 00 .t L-y 0 c 0 5 9 Answer 5: 5 7-y 0 fO Id 3 f f 6. (25 points) Evaluate the integral f + (x + 3 ) 2 dzdydx 12 y z 3 —9—x —9 —x — 2 Hint — Convert to spherical coordinates. 3 L / 5 ZH3 - Answer 6: 6 [ ) 5 ZC[J((Q 7. Find the divergence and curl of the following vector fields. (15 points each) a) i+y F(x,y,z)=x j k 2 +z 7+ (oo, o-O, Divergence 7a): 2 F(x,y,z)yzi+xzj+xyk - I I A1 Ic I L 1 (y (x)I (y-y) 1± (-) O Divergence 7b): Curl 7b):_____ 7 + C Curl7a): b) \/ 0 0 ‘ 1 8. (25 points) Prove that if the curl of a vector field is not zero, then the vector field is not conservative. A i’e/J t f i y/(en4 Cc r y v) F a ‘ {i1I/d ye / /d/ Jr V )y X::: r , / ffl (cc(J, 8 N yd/ fci// To / \/ Vf) a -b r.J tN 0’ AN L r cc N cc SL IN I 1 n - L’J II ,c____ 0 < CD -U Q 0 -0 CD 0. n 0 — x ni