How can we find out the area enclosed by parabola(standard) and latus rectum without using integration?

Question

I managed to solve the question with integration but my teacher won't allow that since it has not been covered in the class.

Answer 1

Hint:

The area under a curve can be computed as the width times the average height.

Consider a right triangle with unit sides. The centroid of the triangle is located at an abscissa which is the average of the abscissas, weighted by the heights. The blue curve shows the product $x^2$ and the abscissa of the centroid is the average of $x^2$ over the average of $x$ (aka the area, which equals $\frac12$).

Hence, as the centroid is known to be located at the $\frac23$ of the basis, we have that the area under the parabola is $\frac13$.