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?

Question

I think I have to use a property but not sure how to solve it from there.

Answer 1

 

 

Integral from 1 to 3 + int. from 3 to 10 = int. 1 to 10 = 4

 

continue ....

 

good luck:)

Answer 2

Think of the intervals of x as line segments being combined to form a single line segment.  So you are combining integrals that cover a local interval to obtain an integral that covers the overall interval.  You will be able to visualize if you draw a coordinate system with x=0 to x=10.  Imagine you are adding segments of the x-axis.

 

If you went from x=1 to x=3, then continued from x=3 to x=10, you covered an interval of x=1 to x=10.

 

 

Therefore,

 

integral from 1 to 3 = (integral from 1 to 10) - (integral from 3 to 10)

 

integral from 1 to 3 = 4 - 7

                             

                             = -3

 

 

This negative integral indicates that a curve from x=1 to x=3 lies below the x-axis.