# Calculus, Chapter 6

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1. General Form of an Antiderivative
G(x) = F(x) + C

C is the constant of integration
2. Notation for Antiderivatives
y = f(x)dx = F(x) + C
3. Initial Conditions and Particlular Solutions
Given: (3x2-1)dx = x3-x+C Passes through (2,4)

F(x) = x3-x+C F(2) = 8-2+C F(2) = 4 when C= -2

So, particular solution: F(x) = x3-x-2
4. Sigma Notation
• n
• ai = a1 + a2 + a3 +...+an
• i=1
5. Important Summation Formulas
Sigma i = [n(n+1)]/2

Sigma i2 = [n(n+1)(2n+1)]/6

Sigma i3 = [n2(n+1)2]/4
6. Lower Sum
(sum of inscribed rectangles)
s(n) = Sigma f(m1)(change x)

mi = 0 + (i-1)(change x)
7. Upper Sum
(sum of circumscribed rectangles)
S(n) = Sigma f(M1)(change x)

Mi = 0+i(change x)
8. Subintervals = change x = [b-a]/n
9. Drefinition of the Area of a Region in the Plane
the area of a region bounded by the graph of f, the x-axis, and the vertical lines x=a and x=b is:

Area = limSigma f(ci)(change x) where (change x)=[b-a]/n
10. Definite Integral
Fundamental Theorem of Calculus
• b
• Sigma f(x)dx = lim Sigma f(ci)(change xi) = F(b)-F(a) = F(x)
• a

a=lower limit; b=upper limit

*remember the line thing...
11. Average Value of a Function
• b
• [1/(b-a)]Sigma f(x)dx = f(C)
• a
12. Mean Value Therorem for Integrals
if f is continuous on the closed interval [a,b] then there exixts a number c in the closed interval [a,b] such that:

• b
• Sigma f(x)dx = f(c)(b-a)
• a
13. Second Fundamental Theorem of Calculus
When we defined the definite integral of f on the interval [a,b] we used the constant b as the upper limit of integration and x as the variable of integration. We now look at a slightly different situation in which the variable x is used as the upper limit of integration.

(d/dx)Sigma dx
14. Guidelines for Integration by Substitution
• 1. Choose a Substitution; choose the inner part of a composite function to sub
• 2. Compute du = g'(x)dx
• 3. Rewrite the integral in terms of the variable u
• 4. Evaluate the resulting integral in terms of u
• 5. Replace u by g(x) to obtain an antiderivative in terms of x