Question

The area bounded by the $X$-axis and the curve $y=4x-x^2-3$ in sq. units is:

• A. $\frac{1}{3}$
• B. $\frac{2}{3}$
• C. $\frac{4}{3}$
• D. $\frac{7}{3}$

Answer 1

The correct option is C $\frac{4}{3}$

$y=4x-x^2-3$
$=-(x^2-4x+3)$
=-(x-3) (x-1)
=(3-x) (x-1)
$\stackrel{3}{\underset{1}{\int }}(4x-x^2-3)dx$
$4\stackrel{3}{\underset{1}{\int }}xdx-\stackrel{3}{\underset{1}{\int }}{x}^{2}dx-3\stackrel{3}{\underset{1}{\int }}1dx$
$=4\left(x^2/2\right){|}_1^3-\left(\frac{x^3}{3}\right){|}_1^3-3\left(x\right){|}_1^3$
$=4(9/2-1/2)-(\frac{27}{3}-1/3)-3(3-1)$
$=16-\left(\frac{26}{3}\right)-6$
$=10-\frac{26}{3}=\frac{4}{3}$