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AP Calculus BC Unit 5 Integrals
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This section covers definite integrals, areas and distances, indefinite integrals, the fundamental theorem of calculus, net change theorem, and the substitution rule.
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Unit 5 | IntegralsSection 5.1 | Areas and Distances
Definition
The area Aof the region S that lies under the graph of the continuous function f is the limit
of the sum of the areas of approximating rectangles:
A = lim Rn = lim [f(x1 )Δx + f(x2 )Δx + ... + f(xn )Δx]
n→∞ n→∞
It can be proved that the limit in the equation above always exists, since we are assuming
that f is continuous. It can also be shown that we get the same value if we use left
endpoints:
A = lim Ln = lim [f(x0 )Δx + f(x1 )Δx + ... + f(xn−1 )Δx]
n→∞ n→∞
In fact, instead of using left endpoints or right endpoints, we could take the height of the ith
rectangle to be the value of f at any number x∗i in the ith subinterval [xi−1 , xi ]. We call the
numbers x∗1 , x∗2 , ..., x∗n the sample endpoints. So a more general expression for the area of
S is
A = lim [f(x∗1 )Δx + f(x∗2 )Δx + ... + f(x∗n )Δx]
n→∞
Unit 5 | Integrals 1
, In general we form lower (and upper) sums by choosing the sample points x∗i so that
f(x∗i )is the minimum (and maximum) value of
f on the ith subinterval.
We often use sigma notation to write sums with many terms more compactly. For instance,
n
∑ f(xi )Δx = f(x1 )Δx + f(x2 )Δx + ... + f(xn )Δx
i=1
So the expression for area can be written as follows:
n
A = lim ∑ f(xi )Δx
n→∞
i=1
n
A = lim ∑ f(xi−1 )Δx
n→∞
i=1
n
A = lim ∑ f(x∗i )Δx
n→∞
i=1
We can also rewrite the formula in the following way
n
n(n − 1)(2n + 1)
∑i =
6
i=1
Section 5.2 | The Definite Integral
Definition of a Definite Integral
If f is a function defined for a ⩽ x ⩽ b, we divide the interval [a, b]into nsubintervals of
equal width Δx = (b − a)/n. We let x0 (= a), x1 , x2 , ..., xn (= b)be the endpoints of these
subintervals and we let x∗1 , x∗2 , ..., x∗n be any sample points in these subintervals, so x∗i lies
in the ith subinterval [xi−1 , xi ]. Then the definite integral of f from ato bis
b n
∫ f(x) dx = lim ∑ f(x∗i )Δx
a n→∞
i=1
provided that this limit exists and gives the same value for all possible choices of sample
points. If it does exist, we say f is integrable on [a, b].
Unit 5 | Integrals 2
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