Question

Find the area of the region bounded by the curves $x=a{t}^2$ and $y=2at$ between the ordinate corresponding to $t=1$ and $t=2$.

Answer 1

Given that $x=a{t}^2....\left(i\right)$

$y=2at....\left(ii\right)\Rightarrow t=\frac{y}{2a}$

putting the value of $t$ in (i), we get $y^2=4ax\cdots \cdots \left(iii\right)$

Putting $t=1$ and $t=2$ in (i), we get $x=a$, and $x=4a$ respectively.

And we need area within these $x$ coordinates.

So, Required area $=2$ area of $ABCD$

$=2{\int }_a^{4a}ydx=2\times 2{\int }_a^{4a}\sqrt{ax}dx\cdots \cdots \left[\text{from (iii)}\right]$

$=4\sqrt{a}{\left|\frac{(x{)}^{\frac{3}{2}}}{\frac{3}{2}}\right|}_a^{4a}$

$=\frac{8\sqrt{a}}{3}[8a^{\frac{3}{2}}-a^{\frac{3}{2}}]=\frac{8\sqrt{a}}{3}\left[7a^{\frac{3}{2}}\right]=\frac{56}{3}{a}^{2}squnits$.