Mohammed Yaaseen Gomdah E WeBWorK assignment due : 03/09/2016 at 03:00am EST. 1. (1 pt) Using disks or washers, nd the volume of the solid obtained by rotating the region bounded by the curves y = 1/x, y = 0, x = 1, and x = 2 about the x axis. Volume = 2. (1 pt) Using disks or washers, nd the volume of the p obtained solid by rotating the region bounded by the curves y = x 1, y = 0, x = 2, and x = 5 about the x-axis. Volume = 3. (1 pt) Using disks or washers, nd the volume of the solid obtained by rotating the region bounded by the curves y = sec(x), y = 1, x = 1, and x = 1 about the x-axis. Volume = 4. (1 pt) Using disks or washers, nd the volume of the solid obtained by rotating the region bounded by the curves y = x2 and y = 4 about the line y = 4. Volume = 5. (1 pt) Which of the following integrals represents the volume of the solid obtained by rotating the region bounded by the curves y = sin(x) and y = 0, with 0 x p about the line y = 1? A. p B. p C. p D. p E. p F. p Z 1 Z0 Z0 1 p Z0 1 Z 0p Z 0p 0 [12 [1 (1 7. (1 pt) Call an improper denite integral type 1 if it is improper because the interval of integration is innite. Call it type 2 if it is improper because the function takes on an innite value within the interval of integration. Classify the type(s) for each of the following improper integrals. Z 0 1 dx ? 1. x2 + 5 Z 2 x ? 2. dx x2 5x + 6 Z0 4 ? 3. x4 e x dx ? 4. Z1 p/2 0 sec(x) dx 8. (1 pt) Consider the integral Z 4x dx 2 1+x If the integral is divergent, type an upper-case \"D\". Otherwise, evaluate the integral. 9. (1 pt) Consider the integral sin(x))2 ] dx sin2 (x)] dx [(1) (1 sin(x))]2 dx [(1) (1 Z 9 sin(x))]2 dx [1 (1 If the integral is divergent, type an upper-case \"D\". Otherwise, evaluate the integral. sin2 (x)] dx [12 1 8 p dx 3 x 9 10. (1 pt) Consider the integral sin(x))2 ] dx Z 4 6. (1 pt) Using disks or washers, nd the volume of the solid obtained by rotating the region bounded by the curves y = x, y = 0, x = 2, and x = 4 about the line x = 1. Volume = 0 4 p dx x If the integral is divergent, type an upper-case \"D\". Otherwise, evaluate the integral. Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 1 Mohammed Yaaseen Gomdah MATH 203 Winter 2016 F WeBWorK assignment number Assignment 7 W14 is due : 03/09/2016 at 03:00am EST. The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information. This le is /conf/snippets/setHeader.pg you can use it as a model for creating les which introduce each problem set. The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind of mistake. Usually you can attempt a problem as many times as you want before the due date. However, if you are having trouble guring out your error, you should consult the book, or ask a fellow student, one of the TA's or your professor for help. Don't spend a lot of time guessing - it's not very efcient or effective. Give 4 or 5 signicant digits for (oating point) numerical answers. For most problems when entering numerical answers, you can if you wish enter elementary expressions such as 2 ^ 3 instead of 8, sin(3 pi/2)instead of -1, e ^ (ln(2)) instead of 2, (2 + tan(3)) (4 sin(5)) ^ 6 7/8 instead of 27620.3413, etc. Here's the list of the functions which WeBWorK understands. You can use the Feedback button on each problem page to send e-mail to the professors. 1. (1 pt) Find dy/dx in terms of x and y if dy dx = The ellipse has two horizontal tangents. The upper one has the equation p p x = 13 y. y= . The right most vertical tangent has the equation x= . That tangent touches the ellipse where y= . Hint: The horizontal tangent is of course characterized by y0 = 0. To nd the vertical tangent use symmetry, or think of x as a function of y, differentiate implicitly, solve for x0 and then set x0 = 0. 2. (1 pt) Find dy/dx in terms of x and y if cos2 (8y) + 2 sin (8y) = y + 3. dy dx = 3. (1 pt) Find the slope of the tangent line to the curve (a lemniscate) 2(x2 + y2 )2 = 25(x2 y2 ) at the point ( 3, 1). m= 4. (1 pt) The graph of the equation 5. (1 pt) For the equation given below, evaluate y0 at the point (2, 2). x2 + xy + y2 = 9 6exy is a slanted ellipse illustrated in this gure: 3x = y + 320. y0 at (2, 2) = 6. (1 pt) Let f (x) = [ln x]8 f 0 (x) = f 0 (e4 ) = 7. (1 pt) Let f (x) = ln[x3 (x + 4)4 (x2 + 2)4 ] f 0 (x) = q d 4 8. (1 pt) Evaluate ln(3 dx q d 4 ln(3 x2 ) at x = 1 = dx Think of y as a function of x. Differentiating implicitly and solving for y0 gives: y0 = . (Your answer will depend on x and y.) 1 x2 ) at x = 1. 9. (1 pt) Let 10. (1 pt) Find y = xlog8 (x) Then dy = dx Note: You must express your answer in terms of natural logs, as Webwork doesn't understand how to evaluate logarithms to other bases. dy = dx dy for each of the following functions dx 5x 15 y = ln p x 8 x2 + 1 y = xcos(x) dy = dx Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 2