This document contains 3 summaries of mathematical concepts: 1) It defines indefinite and definite integrals, and explains that indefinite integrals find antiderivatives while definite integrals calculate the area under a curve between bounds. 2) It states the Mean Value Theorem, which says that if a function is continuous on an interval, there exists a value where the slope of the tangent line equals the average rate of change of the function over the interval. 3) It discusses Riemann sums and defines the Riemann integral as the limit of Riemann sums, which allows approximating the area under a curve through partitions into rectangles.

1. Putting the Pieces Together
From Flickr: The Last Piece by passmeawine
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2. Review: Two Types of Integrals
Indefinite Integral Definite Integral
Verbal Meaning: Verbal Meaning:
Mathematical Definition: Mathematical Definition:
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3. Review: Mean Value Theorem
If
Then
f (x)
a b
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4. 4

5. R6 = ___________________
The value of Rn is ______________,
independent of the value of ___ (number of ________________
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6. By definition of indefinite integral
g(2)­g(1) = g'(c1) = f(c1)
2­1
Height
Slope Slope of
c1 2 c2 3 c3 4 of = of =
1 Riemann
secant line tangent line rectangle
f(c3)
f(c2) By Mean Value theorem
f(c1)
1 2 3 4
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7. 3 & 4.
1 2 3 4
1 2 3 4
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8. 5.
1 2 3 4
1 2 3 4
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