We have
seen “Moment of inertia for
rectangular section”,
“Moment of inertia for the hollow
rectangular section”
and similarly we have also seen “Moment of inertia for
circular section”
and “Moment of inertia for the
hollow circular section" in our previous posts.

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Recently
we were discussing moment of inertia of an area under a curve of given equation about Y axis, now we will see here the method to determine the moment of
inertia of an area under a curve of given equation about axis XX with the help
of this post.

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*Let us see here the moment of inertia of an area under a curve of given equation** about axis XX*

*Let us see here the moment of inertia of an area under a curve of given equation*

*about axis XX*

Let us
consider one curve which equation is parabolic as displayed in following figure
and let us consider that equation of this parabolic curve is as mentioned here.

**x = ky**

^{2}
Let us
determine the moment of inertia of this area about the XX axis. Let us consider
one small strip of thickness dy and at a distance y from the XX axis. We can
also observe here from above figure that x= a, y= b and therefore we can secure
the value of k from above given equation.

a = kb

^{2}
k = a/b

^{2}
Let us
first determine the area of this small strip, dA = (a-x).dy

Area of
this small strip, dA = (a-ky

^{2}).dy
Area of
this small strip, dA = a (1-y

^{2}/b^{2}).dy
Now we
will determine here the moment of inertia of the small strip area about the
axis XX and we can write here as

Moment of
inertia of the small strip area about the axis XX

_{}= y^{2}.dA
Moment of
inertia of the small strip area about the axis XX = y

^{2}. a (1-y^{2}/b^{2}).dy
Moment of
inertia of the small strip area about the axis XX = a (y

^{2}-y^{4}/b^{2}).dy
Now let us integrate the above equation from 0 to b in order to secure the moment of
inertia of this entire area about the axis XX and it is displayed here in
following figure.

Do you
have any suggestions or any amendment required in this post? Please write in
comment box.

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**Reference:**

Strength
of material, By R. K. Bansal

Image
Courtesy: Google

We will
see another important topic i.e.
moment of inertia of uniform thin rod in the category of strength of material, in our
next post.

Very useful derivation in simple steps.

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