Question

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

Short answer

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

Step-by-step solution

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).$

$=64-56+8.$

$=16.$