The integral value of \(\int\limits_4^{20} {f\left( x \right)dx}\) is \(\boxed{\int\limits_4^{20} {f\left( x \right)dx} = 284{\text{ uni}}{{\text{t}}^2}}.\)
Further explanation:
The equation of the line can be expressed as follows,
\(\boxed{\frac{{\left( {y - {y_1}} \right)}}{{{y_2} - {y_1}}} = \frac{{\left( {x - {x_1}} \right)}}{{{x_2} - {x_1}}}}\)
Given:
The value of the function at different values of \(x\) can be written as follows,
\(\begin{aligned}x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4\,\,\,\,\,\,\,\,\,\,\,8\,\,\,\,\,\,\,\,\,\,\,12\,\,\,\,\,\,\,\,\,\,16\,\,\,\,\,\,\,\,\,20\hfill\\f\left(x\right)\,\,\,\,\,\,11\,\,\,\,\,\,\,25\,\,\,\,\,\,\,\,\,\,\,16\,\,\,\,\,\,\,\,\,\,\,9\,\,\,\,\,\,\,\,\,\,\,31 \hfill \\\end{aligned}\)
Explanation:
The function is an increasing function if \({x_1} > {x_2}\), than \(f\left( {{x_1}} \right) > f\left( {{x_2}} \right)\) and the function is decreasing if \({x_1} > {x_2}, {\text{than} f\left( {{x_1}} \right) < f\left( {{x_2}} \right).\)
The \(\int\limits_4^{20} {f\left( x \right)dx}\) can be obtained as follows,
\(\int\limits_4^{20} {f\left( x \right)dx} = \int\limits_4^8 {f\left( x \right)dx} + \int\limits_8^{12} {f\left( x \right)dx} + \int\limits_{12}^{16} {f\left( x \right)dx} + \int\limits_{16}^{20} {f\left( x \right)dx}\)
The equation of line between the points \(\left( {4,11} \right) {\text{and} \left( {8,25} \right)\) can be obtained as follows,
\(\begin{aligned}\frac{{y - 11}}{{25 - 11}} &= \frac{{x - 4}}{{8 - 4}}\\y- 11&=\frac{{14}}{4}\left( {x - 4} \right)\\y&= \frac{{7x}}{2}- 3\\\end{aligned}\)
Similarly the equation between the points \(\left( {8,25} \right) {\text{and} \left( {12,16} \right)\) can be expressed as follows,
\(y = \dfrac{{ - 9x}}{4} + 43\)
The equation between the points \(\left( {12,16} \right) {\text{and} \left( {16,9} \right)\) can be expressed as follows,
\(y = \dfrac{{ - 7x}}{4} + 37\)
The equation between the points \(\left( {16,9} \right) {\text{and} \left( {20,31} \right)\) can be expressed as follows,
\(y = \dfrac{{11x}}{4} - 79\)
The area under the line \(y = \dfrac{{7x}}{2} - 3\) can be obtained as follows,
\(\begin{aligned}\int\limits_4^8 {\left( {\frac{{7x}}{2} - 3} \right)} dx&= \left. {\frac{{7{x^2}}}{4}} \right|_{x = 4}^{x = 8} - \left. {3x} \right|_{x = 4}^{x = 8}\\&= \left( {112 - 24} \right) - \left( {28 - 12} \right)\\&= 72\\\end{aligned}\)
Similarly the area under the line \(y = \dfrac{{ - 9x}}{4} + 43\) can be obtained as follows,
\(\begin{aligned}\int\limits_8^{12} {\left({- \frac{{9x}}{4} + 43} \right)} dx &= \left. { - \frac{{9{x^2}}}{8}} \right|_{x = 8}^{x = 12} + 4\left. {3x} \right|_{x = 8}^{x = 12}\\&= \left( { - 162 + 516} \right) - \left( {72 - 344} \right)\\&= 82\\\end{aligned}\)
The area under the line \(y = \dfrac{{ - 7x}}{4} + 37\) can be obtained as follows,
\(\begin{aligned}\int\limits_{12}^{16} {\left({ - \frac{{7x}}{4} + 37} \right)} dx &= \left.{ - \frac{{7{x^2}}}{8}} \right|_{x = 12}^{x = 16} - \left. {3x} \right|_{x = 12}^{x = 16}\\&= \left( { - 224 + 592} \right) - \left( {126 - 444} \right)\\&= 50\\\end{aligned}\)
The area under the line \(y ={ \dfrac{{11x}}{4} - 79\) can be obtained as follows,
\(\begin{aligned}\int\limits_{16}^{20} {\left( {\dfrac{{11x}}{2} - 79} \right)} dx &= \left. {\dfrac{{11{x^2}}}{4}} \right|_{x = 16}^{x = 20} - \left. {79x} \right|_{x = 16}^{x = 20}\\&= \left( {1100 - 1580} \right) - \left( {704 - 1264} \right)\\&= 80\\\end{aligned}\)
Total area can be obtained as follows,
\(\begin{aligned}\int\limits_4^{20} {f\left( x \right)dx}&= 72 + 82 + 50 + 80\\&= 284{\text{ unit}}{{\text{s}}^2}\\\end{aligned}\)
The integral value of \(\int\limits_4^{20} {f\left( x \right)dx}\) is \(\boxed{\int\limits_4^{20} {f\left( x \right)dx} = 284{\text{ uni}}{{\text{t}}^2}}.\)
Learn more:
1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.
2. Learn more about equation of circle brainly.com/question/1506955.
3. Learn more about range and domain of the function https://brainly.com/question/3412497
Answer details:
Grade: College
Subject: Mathematics
Chapter: Integrals
Keywords: function, decreasing, increasing, monotonic, integral, monotone, data is not monotone, table, sketch, possible, graph, integration, area under the curve.
