exam4_review

exam4_review - MIT OpenCourseWare http:/ocw.mit.edu 18.01...

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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 .
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Lecture 32: Exam 4 Review 18.01 Fall 2006 Exam 4 Review 1. Trig substitution and trig integrals. 2. Partial fractions. 3. Integration by parts. 4. Arc length and surface area of revolution 5. Polar coordinates 6. Area in polar coordinates. Questions from the Students Q: What do we need to know about parametric equations? A: Just keep this formula in mind: ± ² 2 ± ² 2 dx dy ds = + dt dt Example: You’re given x ( t ) = t 4 and y ( t ) = 1 + t Find s (length). ³ ds = (4 t 3 ) 2 + (1) 2 dt Then, integrate with respect to t . Q: Can you quickly review how to do partial fractions? A: When finding partial fractions, first check whether the degree of the numerator is greater than or equal to the degree of the denominator. If so, you first need to do algebraic long- division. If not, then you can split into partial fractions. Example. x 2 + x + 1 ( x 1) 2 ( x + 2) We already know the form of the solution: x 2 + x + 1 A B C = + + ( x 1) 2 ( x + 2) x 1 ( x 1) 2 x + 2 There are two coefficients that are easy to find: B and C . We can find these by the cover-up method. 1 2 + 1 + 1 3 B = = ( x 1) 1 + 2 3 1
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Lecture 32: Exam 4 Review 18.01 Fall 2006 To find C , ( 2) 2 2 + 1 1 C = = ( 2 1) 2 3 ( x → − 2) To find A , one method is to plug in the easiest value of x other than the ones we already used ( x = 1 , 2) . Usually, we use x = 0 . 1 A 1 1 / 3 = + + ( 1) 2 (2) 1 ( 1) 2 2 and then solve to find A .
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This note was uploaded on 01/18/2012 for the course MATH 18.01 taught by Professor Brubaker during the Fall '08 term at MIT.

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exam4_review - MIT OpenCourseWare http:/ocw.mit.edu 18.01...

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