Section 6.5
1
Computing definite integrals
Suppose we wish to compute the total accumulated
change in a given rate function f (t) over the
interval a ≤ t ≤ b. That is, we wish to compute the
definite integral ∫ab f (t)dt .
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One
way
to
do
this
is
to
view the definite integral
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as a specific value of the indefinite integral
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function F(x) = ∫ ax f (t) dt obtained by setting x = b.
But in order to do this, we would like a formula for
F(x).
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€
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We know that F(x) is one of the infinitely many
antiderivative functions that can be found by
applying the integration rules we have identified
(by reversing the differentiation rules). This gives
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∫ f (x)dx = F (x) +C
for some (unknown) value of C. It appears that we
cannot give a complete formula for F(x) since we
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cannot specify
the value of C. But consider what
happens when we compute the difference of the
values of the general antiderivative at x = b and
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x = a: on the one hand we get
(F (b) +C ) − ( F(a) +C ) =€F(b) − F(a)
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Section 6.5
2
while on the other hand,
F(b) − F (a) = ∫ab f (t)dt − ∫aa f (t)dt
= ∫ab f (t)dt − 0
since there is zero area over an interval of zero
length! If we rewrite this last equation in the form
€
b
∫a f (t)dt = F (b) − F(a),
we see that it does not matter what the value of C
is in the antiderivative computation, as it is
€ in the computation of the difference
cancelled
F(b) − F (a). Therefore, to compute the definite
integral ∫ab f (t)dt , find any antiderivative F(x), and
the integral is computed via ∫ab f (t)dt = F (b) − F(a).
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Section 6.5
3
Sum Property for Integrals
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One way to interpret the integral ∫ab f (t)dt is as the
(signed) area under the graph of f over the interval
a ≤ t ≤ b. If we choose an input value c in this
interval so that a < c < b, then it is clear that the
€ integral can be
total area described by this
partitioned into two areas by a vertical line at
x = c . The area to the left of this line is represented
€
by the integral ∫ac f (t)dt , while the area to the right
is represented by ∫cb f (t)dt . It follows that
b
c
b
∫a f (t)dt = ∫ a f (t)dt + ∫c f (t)dt .
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Section 6.5
4
Differences of accumulated changes
Suppose we wish to compute the difference between
total accumulated changes for two rate functions
f (t) and g(t ) over the interval a ≤ t ≤ b. That is, we
wish to compute the quantity
b
€
b
dt − ∫ a g(t)dt .
∫a f (t)€
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These two integrals measure the signed areas
between the horizontal axis and the graphs of the
€ f (t) and g(t ) over the interval a ≤ t ≤ b.
two curves
Their difference, therefore, corresponds to the
signed area between the two curves above the
interval a ≤ t ≤ b.
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€
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But if F(x) is an antiderivative of f (t) and G(x) is
an
€ antiderivative of g(t ), then F(x) − G(x) is an
antiderivative of f (t) − g(t) and we can calculate
€
∫ab f (t)dt
€
−€∫ ab g(t)dt
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€
=€F(b) − F (a) − G(b) − G(a)
(
) (
)
= ( F(b) − G(b)) − ( F(a) − G(b))
= ∫ ( f (t) − g(t))dt
b
a
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Section 6.5
5
Note that if the two curves cross over in the
interval a ≤ t ≤ b, the signs of the areas on either
side of the intersection point are counted as having
opposite sign.
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