5.2 The definite integral  (Page 4/16)

Definition

Let $f\left(x\right)$ be an integrable function defined on an interval $\left[a,b\right].$ Let A ₁ represent the area between $f\left(x\right)$ and the x -axis that lies above the axis and let A ₂ represent the area between $f\left(x\right)$ and the x -axis that lies below the axis. Then, the net signed area between $f\left(x\right)$ and the x -axis is given by

The total area between $f\left(x\right)$ and the x -axis is given by

Rule: properties of the definite integral

1. 
   
   ${\displaystyle {\int }_a^{a}f\left(x\right)dx=0}$
   
   If the limits of integration are the same, the integral is just a line and contains no area.
2. 
   
   ${\displaystyle {\int }_b^{a}f\left(x\right)dx}=\text{−}{\displaystyle {\int }_a^{b}f\left(x\right)dx}$
   
   If the limits are reversed, then place a negative sign in front of the integral.
3. 
   
   ${\displaystyle {\int }_a^b\left[f\left(x\right)+g\left(x\right)\right]dx}={\displaystyle {\int }_a^{b}f\left(x\right)dx}+{\displaystyle {\int }_a^{b}g\left(x\right)dx}$
   
   The integral of a sum is the sum of the integrals.
4. 
   
   ${\displaystyle {\int }_a^b\left[f\left(x\right)-g\left(x\right)\right]dx={\displaystyle {\int }_a^{b}f\left(x\right)dx-{\displaystyle {\int }_a^{b}g\left(x\right)dx}}}$
   
   The integral of a difference is the difference of the integrals.
5. 
   
   ${\displaystyle {\int }_a^{b}cf\left(x\right)dx}=c{\displaystyle {\int }_a^{b}f\left(x\right)}$
   
   for constant c . The integral of the product of a constant and a function is equal to the constant multiplied by the integral of the function.
6. 
   
   ${\displaystyle {\int }_a^{b}f\left(x\right)dx}={\displaystyle {\int }_a^{c}f\left(x\right)dx}+{\displaystyle {\int }_c^{b}f\left(x\right)dx}$
   
   Although this formula normally applies when c is between a and b , the formula holds for all values of a , b , and c , provided $f\left(x\right)$ is integrable on the largest interval.