Let be integrable on and let be a point in that interval

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Let

f

be integrable on

[a,b]

and let

c

be a point in that interval. Define

F = \int_{a}^{c} f(x)\,dx \quad \text{and} \quad G = \int_{c}^{b} f(x)\,dx.

(a) Express

\displaystyle\int_{b}^{a} f(x)\,dx

in terms of

F

and

G

. (b) Prove that

F+G = \displaystyle\int_{a}^{b} f(x)\,dx

for all real numbers

a, b,

and

c

with

a\le c \le b

or

b\le c \le a

. (c) For real numbers

a_1,a_2,\dots,a_n

with

a_{k+1}\ne a_k

and

a_{n+1}=a_1

, prove that

\sum_{k=1}^{n} \int_{a_{k+1}}^{a_k} f(x)\,dx = 0.

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