Question

The expression ${\int }_a^{b}f\left(x\right).dx={\int }_a^{c}f\left(x\right).dx+{\int }_c^{b}f\left(x\right).dx$ is true if and only if a < c < b or a > c > b.

• A. True
• B. False

Answer 1

The correct option is B False

This expression which is given is in fact a property of the definite integrals. Here we are splitting the interval (a, b) to two from a to c and is from c to b. We algebraically add the areas to get the area in the initial interval.
But will that work if ‘c’ is outside the interval (a, b)?
Let’s visualise that. Let’s take is simple curve.

Here in the graph we first take area from a to c and algebraically add area from c to b. But this area from c to b will be having a negative sign since it is on the positive side of y axis and it will cancel out the area outside a to b region. Hence the given statement is false and ‘c’ can lie even outside the a to be region as long as the function is integrable in the concerned regions.