Question

The area of the region $y=ax-b{x}^2$ bounded by $x$-axis in sq. units is

• A. $\frac{a^3}{6b^2}$
• B. $\frac{a^3}{6}$
• C. $a$
• D. $b$

Answer 1

Answer: (A)

$\frac{a^3}{6b^2}$

$y=ax-b{x}^2$
$y=x(a-bx)$
area= $\stackrel{a/b}{\underset{0}{\int }}ydx$
=$\stackrel{a/b}{\underset{0}{\int }}x(a-bx)dx$
$=\stackrel{a/b}{\underset{0}{\int }}ax-b{x}^2=\frac{a{x}^2}{2}{|}_o^{a/b}-\frac{b{x}^3}{3}{|}_o^{a/b}$
$=\frac{a\left(a^2/b^2\right)}{2}-\frac{b\left(a^3/b^3\right)}{3}$
$=a^3/2b^2-\frac{a^3}{3b^2}$
$=\frac{a^3}{6b^2}$