Chapter 3.8, Problem 51PPS

To determine

To find the area of the triangle that is above the $x$ -axis and below the graph of the function $f(x)=-\left|x\right|+10$ .

Answer to Problem 51PPS

  $100$

Explanation of Solution

Given:

Function: $f(x)=-\left|x\right|+10$

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 $f(x)=-\left|x\right|+10$ .

  

From the graph, it can be observed that the vertex of the function is $\left(0,10\right)$ , $x$ -intercepts are -10 and 10.

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 $\left(0,10\right)$ and $\left(10,0\right)$ :

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

  $\begin{array}{l}=\frac{0-10}{10-0}\\ =-1\end{array}$

Equation of the line passing through the points $\left(0,10\right)$ and $\left(10,0\right)$ :

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

  $\begin{array}{l}\Rightarrow \left(y-10\right)=-1\left(x-0\right)\\ \Rightarrow \left(y-10\right)=-x\\ \Rightarrow x+y=10\\ \Rightarrow y=10-x\end{array}$

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

Now,

Area of half triangle is

  $\begin{array}{l}={\displaystyle {\int }_{x=0}^{x=10}{\displaystyle {\int }_{y=0}^{y=10-x}dydx}}\\ ={\displaystyle {\int }_{x=0}^{x=10}\left(10-x\right)dx}\\ ={\left(10x-\frac{x^2}{2}\right)}_0^{10}\\ =10\left(10\right)-\frac{{\left(10\right)}^2}{2}-10\left(0\right)+\frac{{\left(0\right)}^2}{2}\\ =100-50\\ =50\end{array}$

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)

  $\begin{array}{l}=2\times 50\\ =100\end{array}$

Conclusion:

Therefore, the area of the triangle that is above the $x$ -axis and below the graph of the function $f(x)=-\left|x\right|+10$ is $100$ .