Download Integration Techniques & Approximations: Substitution, Parts, Fractions, Rules, Improper I and more Exams Calculus in PDF only on Docsity!
Chapter 7 Integration by Substitution Integration by parts: !
"^ u^ dv =^ uv^ #^ " v^ du
Partial Fractions (see book for a summary of the technique) Trigonometric Identities: ā¢ !
-^^ a^2 "^ x^2 then let x = a sin(t) ! -^^ x^2 "^ a^2 then let^ x = a sec(t) !
^ a^2 +^ x^2 then let x = a tan(t) Tables of Integrals Numerical Approximations of Integrals ā¢ Left(n)
- ā¢ Right(n)Mid(n)
- ā¢ Trap(n)Simp(n) Recall that the Trapezoidal Rule is given by:
!
a^ b^ $^ f^ ( x ) dx^ "^ b^2^ # n^ a^ [^ f^ ( x^0 )^ +^2 f^ ( x^1 )^ +^2 f^ ( x^2 )^ +^ ...+^2 f^ ( xn #^1 )^ +^ f^ ( xn^ )]
Simpsonās Rule is given by:
!
a^ b^ $^ f^ ( x ) dx^ "^ b^^3 # n^ a^ [^ f^ ( x^0 )^ +^4 f^ ( x^1 )^ +^2 f^ ( x^2 )^ +^4 f^ ( x^3 )^ +^ ...+^2 f^ ( xn #^2 )^ +^4 f^ ( xn #^1 )^ +^ f^ ( xn^ )]
There are error estimates for these approximation methods:^ where n is even. For the Trapezoidal rule: !
En^ T^ " ( b^ 12 #^ an )^32 M where M is the maximum value of !
f^ ""( x ) on the interval [a,b]. For Simpsonās rule: !
En^ S^ " ( b 180^ #^ a ) n^54 M where M is the maximum value of !
f (^4 )( x )on the interval [a,b] Improper Integrals ā¢ Evaluate a given improper integral.
- Determine if an improper integral converges or diverges through a comparison test
Chapter 8 Computing Volumes:
- Disc Method: x-axis. Revolve the graph of a continuous non-negative function f on an interval [a,b] over the V =
- Washer Method: axis. Revolve the region between the functions f and g on an interval [a,b] over the x- (Assume f(x) > g(x) > 0)^ V =
- Shell Method: Rotate an area under the function f(x) on an interval [a,b] about the y V = -axis
Lengths of Curves is given by : Let f have a continuous derivative on [a,b]. Then the length L of the graph of f on [a,b]
IF f is the length L of the graph of f on [a,b] is given by given parametrically on [a,b] by differentiable functions and if the velocity v(t) is not 0 on [a,b]. Then
L = a^ b " v ( t ) dt = a^ b "^ # $ % dx dt & ' (^2 + # $ % dy dt & ' (^2 dt
- Determine (symbolically) if the following integrals converge or diverge. Show work, explain.
a. "^ # $ 1 xdx 3 + 1
b. 1 # $ x 3 dx + 1
- Rotate the region bounded by !
Set up the integral, you do NOT have to evaluate the integral. Show work.^^ y^ =^ x^2 +^3 , y = 1, x =^ - 1, and x = 1 around the line y=0. Find the volume.
- Find the length (symbolically) of the parametric curve given by !
Show work.^^ x^ =^ ( t^ +^1 )^32 ,^ y^ =^ ( t^ "^3 )^32 , for^0 ā¤^ t^ ā¤^ 1.
- [20 pts] Find the following integral via trigonometric substitution. Show work.
!
" 16 +^ x^2 dx
