Chapter 3.8, Problem 52PPS

To determine

To find the area of the triangle that is above the $x$ -axis and below the graph of the function $f(x)=-\left|2x\right|+6$ .

Answer to Problem 52PPS

  $18$

Explanation of Solution

Given:

Function: $f(x)=-\left|2x\right|+6$

Formula used:

The area bounded by the curve $y=f(x)$ , the $x$ -axis and the ordinate at $x=a$ and $x=b$ is given by:

  ${\displaystyle {\int }_{x=a}^{x=b}{\displaystyle {\int }_{y=0}^{y=f(x)}}dydx}$

Slope of the line

Calculation:

Consider the graph of the function $f(x)=-\left|2x\right|+6=-2\left|x\right|+6$ .

  

From the graph, it can be observed that the vertex of the function is $\left(0,6\right)$ , $x$ -intercepts are -3 and 3.

Since the graph is symmetrical about $y$ -axis, calculate the area on the positive side of $x$ -axis and multiply the result by 2 to get the total area of the triangle.

Slope of the line passing through the points $\left(0,6\right)$ and $\left(3,0\right)$ :

Slope $=m=\frac{y_2-y_1}{x_2-x_1}$

  $\begin{array}{l}=\frac{0-6}{3-0}\\ =-2\end{array}$

Equation of the line passing through the points $\left(0,6\right)$ and $\left(3,0\right)$ :

  $\Rightarrow \left(y-y_1\right)=m\left(x-x_1\right)$

  $\begin{array}{l}\Rightarrow \left(y-6\right)=-2\left(x-0\right)\\ \Rightarrow \left(y-6\right)=-2x\\ \Rightarrow 2x+y=6\\ \Rightarrow y=6-2x\end{array}$

Total area of triangle $=2\times$ (area of half triangle)

Now,

Area of half triangle is

  $\begin{array}{l}={\displaystyle {\int }_{x=0}^{x=3}{\displaystyle {\int }_{y=0}^{y=6-2x}dydx}}\\ ={\displaystyle {\int }_{x=0}^{x=3}\left(6-2x\right)dx}\\ ={\left(6x-x^2\right)}_0^3\\ =6\left(3\right)-{\left(3\right)}^2-6\left(0\right)+{\left(0\right)}^2\\ =18-9\\ =9\end{array}$

Therefore, total area of triangle that is above the $x$ -axis and below the graph of the function $f(x)$

  $=2\times$ (area of half triangle)

  $\begin{array}{l}=2\times 9\\ =18\end{array}$

Conclusion:

Therefore, the area of the triangle that is above the $x$ -axis and below the graph of the function $f(x)=-\left|2x\right|+6$ is $18$ .