Q. 36 - Calculus

Question

Calculating areas and average values: Express each of the following areas or average values with a definite integral, and then use definite integral formulas to compute the exact value of the area or average value. 

The absolute area between the graph of $f\left(x\right)=3x^2-7x+2$ and the $x$- axis is on $\left[0,4\right].$

Step-by-Step Solution

Answer

The area is $16.$

Step 1. Given information

The given function is, $f\left(x\right)=3x^2-7x+2.$ We need to determine the absolute area between the graph of $f\left(x\right)$ and $x$-axis on $\left[0,4\right].$

Step 2. Calculation

$f\left(x\right)=3x^2-7x+2.$

The absolute area between $f\left(x\right)$ and $x$-axis on $\left[0,4\right]$ is,

$A=\left|{\int }_0^{4}f\left(x\right)dx\right|.$

   $=\left|{\int }_0^4\left(3x^2-7x+2\right)dx\right|.$

    $={\left[\frac{3x^3}{3}-\frac{7x^2}{2}+2x\right]}_0^4.$

    $=\left|{\left(4\right)}^3-\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\left(4\right)}^2+2\left(4\right)+0+0-0\right|.$

     $=\left|69-56+8\right|.$

      $=16.$