Q. 35 - Calculus

Question

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

Step-by-Step Solution

Answer

The area between the graph of $f\left(x\right)$ and $x$ is $16.$

Step 1. Given information

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

Step 2. Calculation

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

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

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

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

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

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

    $=64-56+8.$

     $=16.$