46 - Calculus

Question

$f\left(x\right)=x^2-x-1,g\left(x\right)=5-x^2,[-2,3]$

Step-by-Step Solution

Answer

The exact area between the graphs is $\frac{9}{2}$ or $4.5$ square units.

Step 1: Given information

$f\left(x\right)=x^2-x-1,g\left(x\right)=5-x^2,[-2,3]$

Step 2: Calculation

Consider the following functions:

$f\left(x\right)=x^2;g\left(x\right)=x+2.$

The objective is to find the exact area of the region between the two graphs on the interval $[-3,3]$.

The point of intersection of the two function is,

$$\begin{gathered} x^2=x+2 \\ {x}^2-x-2=0 \\ {x}^2-2x+x-2=0 \\ x(x-2)+1(x-2)=0 \\ (x-2)(x+1)=0 \\ x=-1 or x=2 \end{gathered}$$

The graph is show below, 

The exact area of the region is calculated as follows:

$$\begin{gathered} {\int }_{-1}^2\left|f\right(x)-g(x\left)\right|dx={\int }_{-1}^2\left((x+2)-x^2\right)dx \\ ={\int }_{-1}^2(x+2-x^2)dx \\ ={\left(\frac{x^2}{2}+2x-\frac{x^3}{3}\right)}_{-1}^2 \\ =\left(\frac{2^2}{2}+2\left(2\right)-\frac{2^3}{3}\right)-\left(\frac{(-1{)}^2}{2}+2(-1)-\frac{(-1{)}^3}{3}\right) \end{gathered}$$

$=\frac{9}{2}$ or $4.5$square units.

Therefore, the exact area between the graphs is $\frac{9}{2}$or $4.5$ square units