MAT 266 Lecture 38: Test Review 1

Using an appropriate algebraic substitution, express the polar-coordinates area formula 1/2 [f(theta)]2 d theta so that the angles alpha, beta have unitsof degrees instead of radians Graphs of the functions y = f(x), y = g(x), y = h(x), and y = j(x) in Cartesian coordinates are displayed below. Translate the information provided in these graphs to sketch the polar curves r = f(theta),r = g(theta),r = h(theta), and r = j(theta). Find the area of the "inside loop" of the limacon r = 2 + 4sin(theta). You may want sketch y = 2 + 4sin(x) first. If f is an even function, show that the graph of r = f(theta) is symmetric about the x-axis. Will this ever give rise to a function in the Cartesian sense? Explain.

1. Set up the double integral using polar coordinate to find the area of the following region R: R is bounded by x + y = 1, x + y = 2, z-axis and the y-axis. drdo 2. Set up the triple integral to find the average of f(z, y, z) = zyz in Cartesian coordinates of following region V. . Vis bounded by z =z?+y2 and z = 4-2x2-y2 2π dzdrde

Name MULTIPLE CHOICE·Choose the one alternative that best completes the statement or answers 1) What do the parametric equations x -2 .3t and y -t + 4 describei? A) hyperbola B) a line C) parabola opens to the right D) parabola opens to the left E) ellipse 2) Describe the curve x -3-cos(0, y 2+2 sint) A) ellipse, centre (3, -2) with major axis along the line x 3 B) ellipse, centre (-3, 2) with major axis along the line x--3 C) ellipse, centre (3,-2) with major axis along the line y-2 D) hyperbola, centre (3,-2) with transverse axis along the line x-3 E) hyperbola, centre (3,-2) with transverse axis along the line x-3 3) Find the slope of the curve x = 6t + 3, y = 2t2-7t when t-5. B) 0 C) 6 D) 3 4) Find the slope of the curve x=5cos t,y-3sin t at t=π A) B) 1 5) Find the rectangular coordinates of the point with polar coordinates P B)(2, C)(3,1) D)65,2) 6) Convert the point with Cartesian coordinates (+1, -1) to polar coordinates. 7) Convert x2 +y2 = xy to polar coordinates.