MAT 266 Lecture Notes - Lecture 35: Polar Coordinate System, Pythagorean Theorem

Show all your work for full credit. The textbook, lecture notes, and elec- tronic devices with symbolic computing capacity are NOT allowed 1. Evaluate cos(x + 2y)dA, whe 2. Evaluate the following double integral Jo Jz 3. Evaluate the double integrals by converting them to polar coordinates. 2心ー昏 4. Evaluate the double integrals by converting them to polar coordinates. 5. Evaluate the triple integrals dxdydz, 2 drdzdy, 3. rdzdrd9. 6. Find the volume between the graph of the function f(r, y) - zy and the rect- angle R = {(z, y)1-2 3and-1 7. Evaluate the following double integrals y 5) 0 z+2 3. 2ded 8. Let S be the triangular region with vertices (0,0), (-1, 1), and (1, 1). (Sketch the region first!) . Find the centroid ofS . If the density at each point in S is proportional to the point's distance from the y-axis, find the mass of S.

4. Without eliminating the parameter, do the following for the curve defined by 1 = sin 2t, y = cos 2t· . (a) Find dy and dry. (b) Evaluate and瓜at (c) Use the second derivative test to determine if the polar curve has a relative maximum, and at t = 0. dz2 minimum, or neither at the point where t 0。JYA. 5. Find the length of the curve defined by r = 2 + 3t2, y 1 + 2A for 0-t-3. (Hint factor the radicand (inside the square root) so that a u-substitution will work.) 6. In the polar coordinate system provided (separately), plot each of the points whose polar coordinates are given. - 4, D-3, , B

Sample ⠣ Find the interval of convergence forZr-1)-x-- Do not test the end points 1. 3" 2. Find a power series for the functioncentered at c 1 3. Use the defnition (Taylors formula) to find the series centered at c - for () Do not find the interval of convergence. 4. Find the corresponding rectangular equation represented by the parametric equations 2 + cos θ and y = 2 + 5 sin θ by eliminating the parameter. x Find the ArcLength of the curve x=arcsint,yslnf-2, on the interval 5. 6. Find two sets of polar coordinates with 0 θ 2π for the point with rectangular coordinates 2,v2 7. Find the equation of the tangent line for the curve given by x-+2 and y--2t at the point where t =-5. Convert the polar equations to rectangular form. 8. 2π