MOMENT OF INERTIA OF AREA UNDER A CURVE OF GIVEN EQUATION ABOUT X AXIS

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.

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.

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 = ky²

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 = kb²

k = a/b²

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

Area of this small strip, dA = (a-ky²).dy

Area of this small strip, dA = a (1-y²/b²).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 = y².dA

Moment of inertia of the small strip area about the axis XX = y². a (1-y²/b²).dy

Moment of inertia of the small strip area about the axis XX = a (y²-y⁴/b²).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.