Q. 38 - Calculus

Question

Calculate 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 area between the graphs of $f\left(x\right)=4-x^2.$ and $1-2x$ on $\left[-4,4\right].$ 

Step-by-Step Solution

Answer

The area between $f\left(x\right)$ and $1-2x$ is $24.$

Step 1. Given information

The given function is $f\left(x\right)=4-x^2,$we need to determine the area between the graph of $f\left(x\right)$ and $1-2x$ on $\left[-4,4\right].$

Step 2. Calculation

The area between $f\left(x\right)$ and $1-2x$ on $\left[-4,4\right]$ is,

$\left|{\int }_{-4}^4\left[\left(4-x^2\right)-\left(1-2x\right)\right]dx\right|.$

$={\int }_{-4}^0\left(3-x^2+2x\right)dx+{\int }_0^4\left(3-x^2+2x\right)dx.$

$={\left[3x-\frac{x^3}{3}+x^2\right]}_{-4}^0+{\left[3x-\frac{x^3}{3}+x^2\right]}_0^4.$

$=\left[0-0+0-3\left(-4\right)+\frac{{\left(4\right)}^3}{3}+{\left(-4\right)}^2\right]+\left[3\left(4\right)-\frac{{\left(4\right)}^3}{3}+{\left(4\right)}^2\right].$

$=12+\raisebox{1ex}{$64$}\!\left/ \!\raisebox{-1ex}{$3$}\right.-16+12-\raisebox{1ex}{$64$}\!\left/ \!\raisebox{-1ex}{$3$}\right.+16.$

$=24.$