Question

Graph the given curve and find the area of a bounded region.

y= 4x - x^2

Answer 1

Solution

The required area is found by using the formula

$A = \int\limits_a^b {f\left( x \right)} \,\,dx$

Here given that

$f(x)= 4x - x^2$

When plotted as shown below, we can see, the bounded area, in this case, is under the curve $f(x)= 4x - x^2$, the $x-axis$ and the two limits $x=0$ and $x=4$

Therefore, the required area is calculated as

$A = \int\limits_0^4 {(4x-x^2)} \,\,dx$

$\begin{array}{l} A = \int\limits_0^4 {\left( {4x - {x^2}} \right)} \,\,dx\\ A = \left[ {\frac{{4{x^2}}}{2} - \frac{{{x^3}}}{3}} \right]_0^4\\ A = \left[ {\left( {2{{\left( 4 \right)}^2} - \frac{{{{\left( 4 \right)}^3}}}{3}} \right) - \left( {2{{\left( 0 \right)}^2} - \frac{{{{\left( 0 \right)}^3}}}{3}} \right)} \right]\\ A = \left[ {32 - \frac{{64}}{3} - 0 + 0} \right]\\ A = \frac{{32}}{3} \end{array}$

Hence required area is $\frac{32}{3}$ square units