MAT 127 Final: S08finalsol

• School: University of Maine
• Department: Mathematics and Statistics
• Course: MAT 127
• Professor: All

Document Summary

Part i: (20 pts) compute: (a) z x x2 + 5 dx. Easy solution: let u = x2 + 5, du = 2x dx. Let x = 5 tan , dx = 5 sec2 d . 5z 5 tan p5 tan2 + 5 sec2 d = 53/2z tan sec3 d . Let u = sec , du = sec tan d : , and we get the same result as before. (b) z sin5(x) cos4(x) dx. Z sin4(x) cos4(x) sin(x)dx = z (sin2(x))2 cos4(x) sin(x)dx. Letting u = cos(x), du = sin(x)dx, this is. + c: (20 pts) a cone is formed by rotating the shaded region around the y-axis: y = 5 . Use calculus to nd the volume of the cone. Using dx: a thin vertical rectangle of width dx creates a cylindrical shell when rotated around the y-axis. The height of the shell is 5 5x/3, and the radius is x.