MATH 201- Midterm Exam Guide - Comprehensive Notes for the exam ( 42 pages long!)

8. For each of the following iterated integrals, do the following: Sketch the region of integration. Where applicable, give a geometric interpretation of the iterated integral. Evaluate each double integral. Change the order of integration where nec- essary. (a) 104 fets dydr (b)//쁩udy dr (c)//gudy @i jsdady (h)? fsinrardy (i).t{r)lydydz 0 ev 1 2 dzd @j feë´ drdy((o) 2 4x c s(y ) dydr (p)//VI阿drdy a) J sin(3戸) dydx (r) fsin(H) dzdy (s)// dady (t) 9 3 sin (㎡) drdy (u) 4 2ye'dydr (v)//e-va dydz 0 y/2 1 Vy vz 0 z ivITpdrdy@i 2向dady (y)..-75dy dr (w) (1 3)

o Math Qulz 15: 15.8-16.3 (Take-Home) All work must be clearly shown for partial credit 1. Evaluate the integral by changing to cylindrical coordinates. (5 pts) x2+y dz dy dr 2. Consider the transformation T: x -2u+3v y 3u+2v a) Sketch the image of S under T of the unit square R-{(u,v) 0susl, Osvs in the uv- plane using the change of coordinates. b) Evaluate (3x+ 2y A (5 pts) 3. Evaluate Jx'yd -xdy where C is the circlex+y-1 with counterclockwise orientation. (5 pts 4. Evaluate F-dr where F(x,ya) (+ 3yi+3j+ xyk and C is given by (5 pts 5. Show that F(x.y) - sin yi + (xcos y + sin y)j is a conservative vector field and find a function f such that F -vf (5 pts 8. Show that F(x.y) (2x+y'+3'yi+ (2xy'+3y')i is a conservative vector field and use his fact to evaluate IF-dr where C sthe arc of the an er(t)-(vsin) Ostsπ (5 pts

Ma 201 Spring 2013 Test 4 Show all work clearly and circle your final answers. Partial credit will only be given if I can follow what steps you were taking for the problem. Use the back of the page if needed. Explain the use of triple integrals and the general method of how they are set up. Mark as True (T) or false(F): f(x, y) dA = dV for some curve z = f (x, y) J (x, y) = 1 /J(u, v) Double integrals can be used tor area The projection of spherical coordinates onto a plane are cylindrical coordinates Solve; x2dA where the region is bounded between y = cos(x) and y = sm(x) where cos(x) le yle sin (x) within the first 2 pi