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CALCULUS 1 PROBLEMS & SOLUTIONS
6.2.2
Properties Of The Definite Integral
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Review
1. Integrability Of Continuous Functions
Theorem
1
¾¾¾¾¾¾¾¾¾¾¾¾
¾¾¾¾¾¾¾¾¾¾¾¾
The proof of this theorem involves subtle properties of the real number system
derived from its
completeness property and consequently is beyond the scope of this site.
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2. Properties Of The Definite Integral
Theorem
2
¾¾¾¾¾¾¾¾¾¾¾¾
Suppose f and g are
continuous (hence integrable) on an interval containing the points a,
b,
and c,
where a < b
and c can be anywhere in the interval,
and let k be a constant. Then:
¾¾¾¾¾¾¾¾¾¾¾¾
Proof
i. Since f
+ g
is continuous on [a, b], by Theorem
1 it's integrable there, and:
ii. Similar to part i.
iii. This follows from the fact
that all the Riemann sums are 0, which itself follows from the fact that
all the rectangles in all the
Riemann sums have bases of length 0.
EOP
Remark 1
Property vii is a generalization of the triangle inequality |x
+ y|
£
|x|
+ |y|.
For a visual help on the
inequality case, refer to Fig. 1, where we assume A1 > A2. The left-hand
side of Property vii is
|A1 – A2| = A1 – A2, while its right-hand
side is A1 + A2, because we take the absolute
value before
we integrate (as if we flip the region measured by A2 upward about
the x-axis before we integrate).
Since A2 > 0, we have A1 – A2 < A1 + A2.
|
|
Fig. 1
|
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Problems & Solutions
Solution
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2. Suppose f
and g are continuous functions, f
is even, g is odd, and a is a
constant. Prove that:
Solution
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