MAT235Y1- Final Exam Guide - Comprehensive Notes for the exam ( 35 pages long!)

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.

Please read the directions for each problem carefully. You may work together and receive help on this exam; however, each student must turn in his or her own exam. This exam is due by 11:30am on Monday November 27th. Late exams will not be accepted. If you are unable to attend class on Monday November 27th, you must scan and email the exam to me by 11:30am (11 points) Given the function f(x) =- (a) Find the domain of the function in interval notation. (x-1)2 x2+1 I. Answer (in interval notation):0o o (b) Find the intercepts of the function. All work must be shown. Write your answers as ordered pairs. o Answer(s) (written as ordered pairs): L101-10.1) (c) Find the equations of any asymptotes. All work must be shown. L91 C110) Answer(s) (written as equations) (d) Find the critical numbers. All work must be shown

Iul, iis il tiins page in the space provided. Each problem is worth 20 points. Note: This is a take-home exam. You may use your textbook and/or work with your classmates on these problems, but the work you submit for grading MUST be your own. Exams are due no later than 5:00 pm on Tuesday, December 42 I. The box shown has dimensions: a bc 6 ft. The temperature at any point within the box is given by the formula: T T(x, y, z) x2 +4yz. Compute the average temperature within the box by evaluating a triple integral in Cartesian coordinates. The center of pressure of a vertical sail on a sailboat is the point (tpyp) at which the total aerodynamic force on the sail may be assumed to act. If the sail is represented by a plane region D, then the center of pressure is given by the formula 2. Parts Of The Mainsail Baten Topping Litt Reefing Use this formula to find the center of pressure on a sail represented by the triangular region D in the xy-plane with vertices (units are meters): Cunningham