1. [SOLVED] Integral problem

If g(x) is continuous for all x, which of the following integrals necessarily hv the same value?

Solved

I think that 1 and 2 won't have the same value, but I am not completely sure. I have no idea when comparing 1 and 2 to 3 and 4.

2. Originally Posted by solars
If g(x) is continuous for all x, which of the following integrals necessarily hv the same value?

1.)integral sign (a to b) f(x)dx
2.)integral sign (a to b) abs(f(x))dx
3.)integral sign ((a-c) to (b-c)) f(x+c) dx
4.)integral sign (a to b) (f(x)+c)

A 1, 2 only
B 1, 3 only
C 1, 2, 4 only
D 2,3,4 only
E no two necessariyt hv the same value

I think that 1 and 2 won't have the same value, but I am not completely sure. I have no idea when comparing 1 and 2 to 3 and 4.
For 3. substitute u = x + c. Then:

$\int_{a-c}^{b-c} f(x + c) \, dx = \int_{a}^{b} f(u) \, du$ which is actually equivalent to $\int_{a}^{b} f(x) \, dx$ since a definite integral does not depend on the (dummy) variable used.

The correct answer is therefore option B.

To show that 1. and 2. are not equal, a simple counter-example is sufficient. Consider f(x) = -x, a = 0 and b = 1.

For 4. note that

$\int_{a}^{b} f(x) + c \, dx = \int_{a}^{b} f(x) \, dx + \int_{a}^{b}c \, dx = \int_{a}^{b} f(x)\, dx + c(b - a) \neq \int_{a}^{b} f(x)\, dx$ unless a = b or c = 0.