Math 414: Analysis I
Integration Problems
1. Suppose that f : [a, b] → R is a monotone increasing function. That is, for each
x ≤ y, f (x) ≤ f (y), for all x, y ∈ [a, b]. Prove that f is Riemann integrable on [a, b].
You may use the following facts without proof:
• For each > 0, there is a k > 0 such that k [f (b) − f (a)] < .
• There is a partition P of [a, b] such that (xi − xi−1 ) < k for all i = 1, 2, . . . , n.
2. Suppose that f : [a, b] → R is continuous. Prove that there is a c ∈ [a, b], such
Rb
that a f dx = f (c)(b − a). Remark: This is known as the Mean Value Theorem for
Integrals.
3. Suppose that f and g are differentiable on [a, b] and that f 0 and g 0 are continuous on
[a, b]. Prove that f 0 g and g 0 f are integrable on [a, b] and that from the Fundamental
Theorem of Calculus we obtain
Z
b
Z
0
f g dx = f (b)g(b) − f (a)g(a) −
b
g 0 f dx.
a
a
Remark: This is the integration by parts formula.
4. Suppose that f : [a, b] → R has the property that f (x) = 0 for all x 6= x0 , and
f (x0 ) = c for some c 6= 0. Prove that f is Riemann integrable on [a, b] and comRb
pute a f dx. Remark: Via induction, you could extend this result to include those
functions f that are non-zero on a finite number of points in [a, b].
5. Suppose that f : [a, b] → R is integrable. Prove that
Z
m(b − a) ≤
b
f dx ≤ M (b − a)
a
for some m, M ∈ R.
6. Suppose that f : [a, b] → R is continuous and that f (x) ≥ 0 for all x ∈ [a, b]. Prove
Rb
that if a f dx = 0 then f (x) = 0 for all x ∈ [a, b]. Suggestion: Use contradiction
and recall Homework 9 Problem 4.
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