Question

If the area of the region bounded by the curves, y = x2 , y = 1 x and the lines y = 0 and x = t (t > 1) is 1 sq. Unit, then t is equal to

Answer 1

Solution:

The given curves are

y= x², y= 1 x and the line y=0 and x=t (t>1).

The point of intersection of y=x² and y= 1 x is

x= x²

→ x=0,1

Gives two point of intersection (0,0) and (1,1).

Considering , the line given x=t, t>1

And finding it's area→→ Area under the Parabola from 1 to t - Area under the line starting from 1 to t= 1 square unit

$\int\limits^k_1 {x^2}  \, dx-\int\limits^t_1 {x} \, dx  = 1, {\text{here}, k= t^2$

$[\frac{x^3}{3}]\left \{ {{x=t^2} \atop {x=1}} \right. - \frac{x^2}{2}\left \{ {{x=t} \atop {x=1}} \right. =1\\\\ \frac{t^6}{3} -\frac{1}{3}-\frac{t^2}{2}+\frac{1}{2}=1 \\\\ \frac{t^6}{3}- \frac{t^2}{2}=\frac{5}{6}\\\\ 2 t^6 - 3 t^2=5$

→t= 1.33 is the solution of the area of the region bounded by the curves, y = x² , y = 1 x and the lines y = 0 and x = t (t > 1) is 1 sq. Unit.