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

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