Question

I: The area bounded by the line $y=x$ and the curve $y=x^3$ is $1{/}_2$ sq. units.
II: The area bounded by the curves $y=x^3$ and $y=x^2$and the ordinates $x=1$, $x=2$ is $\frac{7}{12}$ sq. units.
Which of the above statement is correct?

• A. Only I
• B. Only II
• C. Both I and II
• D. Neither I nor II.

Answer 1

The correct option is A Only I

$SI$: $\stackrel{1}{\underset{0}{\int }}(x-x^3)dx=(\frac{x^2}{2}-\frac{x^4}{4}){|}_0^1$
$=2(\frac{1}{2}-\frac{1}{4})=2\left(1/4\right)squnits$
$=\frac{1}{2}squnits.$
$SII:$ $\stackrel{2}{\underset{1}{\int }}(x^3-x^2)dx$
$=\frac{x^4}{4}-\frac{x^3}{3}{|}_1^2$
$=(\frac{2^4}{4}-\frac{2^3}{3})-(\frac{1}{4}-\frac{1}{3})$
$=\frac{15}{4}-\frac{7}{3}=\frac{45-28}{12}=\frac{17}{12}squnits.$

only $I$ is true