find the moment of inertia of a rotating body around the y axis, with the density of 1

[subtle stag]

I_y=\int_{A}\rho \cdot x^2 dA

where A is the shaded area

and dA=dxdy

Now we must find the boundaries of x and y

[subtle stag]

So when y\in[0,2], the variable x varies from x=0 to the line that passes through the points (2,0) and (4,4) which is y=2x-4 after solving for x, the upper bound is (y+4)/2

When y \in [2,4] the variable x varies from line that passes through the points (0,2) and (4,4) to the line y=2x-4 which implies that x \in [(y-2)/2;(y+4)/2]

Iy=\int_{0}^{2}\int_{0}^{(y+4)/2}1 \cdot x^2 dxdy+\int_{2}^{4}\int_{(y-2)/2}^{(y+4)/2}1 \cdot x^2 dxdy

Solve both integrals

[wise pond]

thank you!

!solved

.solved

[lean leaf]

Solved

Post marked as solved by @wise pond.

Use .unsolved if this was a mistake.

[subtle stag]

U are welcome bro!

[subtle stag]

If you need any help dm me

[wise pond]

deal