Let ∫ [8.5 1]f(x)dx=6, ∫ [3.5 1]f(x)dx=6, ∫ [8.5 6]f(x)dx=3. Find ∫ [6 3.5]f(x)dx=? and ∫ [6 3.5](6f(x)-6)dx=?

Answer 1

The solution to this problem is based on the following formula.

so int(from 1 to 8.5) = int(from 1 to 3.5) + int(from 3.5 to 6) + int(from 6 to 8.5) which according to the given values results in:

6 = 6 + int(from 3.5 to 6) + 3 to solve for int(from 3.5 to 6 ) = -3;

Note that in the above, I have dropped f(x)dx term for simplicity.

Next, we solve int(from 3.5 to 6 of (6f(x) - 6)dx = 6int(from 3.5 to 6) - 6(int from 3.5 to 6 dx) which is equal to (6*-3) - 6 int(from 3.5 to 6) dx = -18 - 6*(6 - 3.5) = -18 - 15 = -33;