If the integral from 1 to 10 f (x) dx=4 and integral from 3 to 10 f (x) dx=7 , then what is the integral from 1 to 3 f (x) dx? I think I have to use a property but not sure how to solve it from there.

Question

What is the answer?

Answer 1

The integral is simply the area under a curve. So the area from x=1 to 10 is 4 and the area from x=3 to10, equalling 7, is included in the former area. The curve lies above and below the x axis. The difference between these areas is 4-7=-3, suggesting that the net area of f(x) for x between 1 and 3 is below the x axis. Area between 1 and 10=(area between 1 and 3)+(area between 3 and 10). So 4=(area between 1 and 3)+7.