{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 0309 - Math 31A 2010.03.09 MATH 31A DISCUSSION JED YANG 1...

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

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 π 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 u du = 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 x dx. Show that these yield different answers and explain the apparent contradiction.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}