Practice Questions

Practice Questions - MIT OpenCourseWare http://ocw.mit.edu...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.01 Practice Questions for Exam 4 – Fall 2006 Problem 1. Problem 2. Evaluate � Evaluate � x−4 dx. (x + 1)(x2 + 4) 2 0 (x2 dx by making the substitution x = 2 tan u. + 4)2 Problem 3. a) Derive a reduction formula relating �1 � 2 b) Let F (x) = e−x dx. Express 0 � 1 2n −x2 xe 0 dx to � 1 2 x2n−2 e−x dx. 0 1 2 −x2 xe dx in terms of values of F (x). 0 Problem 4. Find the volume of the solid obtained by rotating about the y -axis the finite region bounded by the positive x- and y -axes and the graph of y = cos x. Problem 5. Make a reasonable sketch of one loop of the polar curve r = sin 3�, and find the area inside it. Problem 6. Let x(t) = cos3 t, y (t) = sin3 t, 0 � t � � /2 be a parametric representation of a curve. a) Compute the arclength of the curve. b) Compute the surface area of the surface formed by rotating the curve around the x-axis. Problem 7. Set up an integral for the length of one arch of the curve y = sin x, and by � estimating the integral, tell how this length compares with � 2. Problem 8. A circular metal disc of radius a has a non-constant density � (units: gms/cm2 ); the density at a point P on the disc is given by � = r 2 , where r is the distance of the point from the center of the disc. Set up and evaluate a definite integral giving the total mass of the disc. Problem 9. a) Sketch the curve given in polar coordinates by r = 1 + cos � b) Find the polar coordinates of the following two points (show work): (i) where the curve in part (a) intersects the circle of radius 3/2 centered at the origin; (ii) where the above curve intersects the circle of radius 3/2 centered at the point x = 3/2 on the x-axis. Other kinds of problems: Other kinds of partial fractions decompositions; sketching curves given parametrically, finding their arclength; finding surface area for rotated curves in xy -coordinates; deriving polar equations of curves given geometrically, changing from rectangular equa­ tions to polar and vice-versa. ...
View Full Document

This note was uploaded on 10/10/2011 for the course MATH 31 taught by Professor Blake during the Spring '11 term at MIT.

Page1 / 2

Practice Questions - MIT OpenCourseWare http://ocw.mit.edu...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online