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Unformatted text preview: Math 31A 2010.03.09 MATH 31A DISCUSSION JED YANG 1. Theorems 1.1. The Substitution Method. If F ( x ) = f ( x ), then integraldisplay f ( u ( x )) u ( x ) dx = F ( u ( x )) + C. 1.2. Area Between Graphs. If f ( x ) g ( x ) on [ a,b ], then the area between the graphs is integraltext b a f ( x )- g ( x ) dx . 1.3. Volume of a Solid. If the cross section of a solid between [ a,b ] has area A ( x ), then the volume is integraltext b a A ( x ) dx . 1.4. Solid of Revolution. If a solid of revolution is formed with radii r < R on [ a,b ], then the volume is integraltext b a ( R 2- r 2 ) dx . 2. More Topics in Integration 2.1. Exercise 5.6.50. Evaluate the indefinite integral integraldisplay cos x x dx. Solution. Let u = x , then du = 1 2 x dx . So integraltext cos x x dx = integraltext 2 cos udu = 2 sin u + C = 2 sin x + C . square 2.2. Exercise 5.6.54. Can They Both Be Right? Use u = tan x and u = sec x to evaluate integraldisplay tan x sec 2 xdx....
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This note was uploaded on 09/19/2011 for the course MATH 31A taught by Professor Jonathanrogawski during the Winter '07 term at UCLA.
- Winter '07
- Substitution Method