Calculus 5.2

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4.3 Reimann Sums Definite Integrals
Greg Kelly, Hanford High School, Richland,
Washington
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When we find the area under a curve by adding
rectangles, the answer is called a Rieman sum.
The width of a rectangle is called a subinterval.
The entire interval is called the partition.
subinterval
partition
Subintervals do not all have to be the same size.
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If we use subintervals of equal length, then the
length of a subinterval is
The definite integral is then given by
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Leibniz introduced a simpler notation for the
definite integral
Note that the very small change in x becomes dx.
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upper limit of integration
Integration Symbol
integrand
variable of integration (dummy variable)
lower limit of integration
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We have the notation for integration, but we
still need to learn how to evaluate the
integral. If we can evaluate the integral, the
function is integrable.
Continuity implies integrability!
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Note that this is continuous over all reals, so
we can integrate it.
Ex. 1 Evaluate
Lets use subintervals of equal width
Hey! This is our upper bound!
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Note While the quantity area is a positive
value, definite integrals can be negative. What
do you think this means in terms of the area??
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Ex. 2 Using limits, evaluate
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