# How to determine the same moment of inertia in two different ways?

• Tapias5000
In summary, the conversation discusses a problem involving finding the moment of inertia of an area with respect to the y-axis using different rectangular elements and two different methods. The relevant equation for determining the moment of inertia is shown, but it is noted that x should be squared for y-axis rotation inertia. The two methods are then shown.

#### Tapias5000

Homework Statement
determine the moment of inertia of the area with respect to the y-axis. with different rectangular elements, solve the problem in two ways: (a) with thickness dx, and (b) with thickness dy.
Relevant Equations
## I_x=\int _{ }^{ }y^2dA ##

My solution is

now I am asked for the same result but in this form but I don't know where to start.

$$2\sigma \int_0^1 x^2 y \ \ dx=32\sigma \int_0^{\pi/2} \sin^2\theta \cos^2\theta d\theta$$
where ##\sigma## is area mass density of board.

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anuttarasammyak said:
$$2\sigma \int_0^1 x^2 y \ \ dx=32\sigma \int_0^{\pi/2} \sin^2\theta \cos^2\theta d\theta$$
where ##\sigma## is area mass density of board.
waat, Can you tell me the name of this method?

I misinterpreted y so
$$2\sigma \int_0^1 x^2 y \ \ dx=2\int_0^1 x^2 (4-4x^2) \ \ dx$$
This a way (b).
Tapias5000 said:
Homework Statement:: determine the moment of inertia of the area with respect to the y-axis. with different rectangular elements, solve the problem in two ways: (a) with thickness dx, and (b) with thickness dy.
Relevant Equations:: ## I_x=\int _{ }^{ }y^2dA ##
This relevant equation seems inappropriate because x should be squared for y-axis rotation inertia.
For (a)

$$2\sigma\int_0^4 dy \int_0^{\frac{\sqrt{4-y}}{2}} x^2 dx$$

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