MATH 201 Final, Fall 2012

Math 201 Final Exam (Fall 2012) li. Yamani Probiém 1 (answer on page { of the booklet) (18 points) Let R be the region in the first quadrant bounded by the x-axis, the y-axis and the ci rele x? +(y- 1p =e Let € be the boundary of R traced counterclockwise. Use Green’s theorem to find ihe outward flux of the ficld F= ( + 3x) i+ ( yte "cos x} j Problem 2 (answer on pages 2,3 of the booklet.) (18 points) ow that the differential form in the following integral is exact, and then evaluate the integral. (2.1.0) | (ye 4 2 are{ aoe + 2 dy+ (xe! +xe \ic yo +l (1.0.0) Problem 3 (answer on page 4 of the booklet.) (18 points) Find the flux done by the vector fidd F=xit+yj+k over the curve C: x=y° from (1, 1) to 0, 0). Problem 4 (answer on page 5 of the booklet) (18 points) y Integrate Sls, y) == suit See over the curve C: y = x +41 from (0, 1) to (1, 2). yi6x® +1 Probitm 5 (answer on pages 6, 7 of the booklet) (18 points) ‘ : 2 Let & be the ellipse x? — xp t y =% Use the transformation x= J2u- iF: ¥ ¥= [2u+ fs to rewrite J] (2 —xy+ y? dale, y) as an integral over an appropriate region Gin the uv-plane. Then evaluate the ?°———— uv -integral over G. problem 6 (answer on pages 8, 9 of the booklet) (6 points each) Write the volume of the region in the first octant bounded by the planes x = 0,7 =0,2 =0 and the planes xez=lyptac=2,asan iterated triple integral using the order of integration (a) dxdydz (b) dzdydx (c) Evaluate the inte gral using the order dxdyelz Problem 7 (answer on pages 10, 11 of the booklet) : (9 points each) Let D be the region bounded above by the sphere x+y + = =2 and below by the cone gaal4 (10 pts. each) Set up the triple integrals in cylindrical coordinates that give the volume of D using the order of integration (a) dzdrd@ (b) drdzdO Problem 8 (answer on pages 11, 12 of the booklet) (9 points each) Let D be the region bounded above by the sphere x? + y 422 =4 and below by z =1 (10 a a up the triple integrals in spherical coordinates that give the volume of D using the order of d, integration pd gd @ (b) dédpdé Problem 9 ( answer on page 13 of the booklet) (18 points) Find the absolute maximum and minimum values off (x.y. zj= xy" =” on the sphere + y +2 =3., Problem 10 ( answer on page 14 of the booklet). (9 points each) Which of the following series converge, and which diverge? a (5° +n 20 1 (a) = bye a n=t eit m3 (Inn) a 11 (answer on pages 15, 16 of the booklet) (20 points) Consider f(x) = ln(l + x) Find the Taylor polynomial p(x) generated oy (x)= inl +x) about x =0- Approximate In(1.1) by pol?) and estimate the resulting error. Also Approximate in(0.9) by ps (?) and estimate the resulting error.