Thread: Optimization or area under parabola

This question isn't really optimization but the second part is (which I know how to do). Its the first part i'm stuck at.

The cross-section of a building constructed inside a parabolic framework is shown in the diagram.

(sorry for bad drawing)

a) Given that PQ is 2xm show that the shaded area is 6x - 2x^3

The only thing I can think of is to multiply the equation of the parabola
(3 - x^2) by 2x, which gives 6x - 2x^3 .

Can you tell how the proper way to do it is?

Thanks

Originally Posted by r_maths

This question isn't really optimization but the second part is (which I know how to do). Its the first part i'm stuck at.

The cross-section of a building constructed inside a parabolic framework is shown in the diagram.

(sorry for bad drawing)

a) Given that PQ is 2xm show that the shaded area is 6x - 2x^3

The only thing I can think of is to multiply the equation of the parabola
(3 - x^2) by 2x, which gives 6x - 2x^3 .

Can you tell how the proper way to do it is?

Thanks

The area of the shaded area is the width of the rectangle (which is 2x)
times the height of the rectangle. But the height is 3-x^2, so the area is

2x [3 - x^2) = 6x - 2 x^3.

RonL