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Finding a centroid

Find the centroid of the region under the curve y = e x over the interval 1 x 3 (see the following figure).

On the x y plane the curve y = e to the x is shown from x = 0 to x = 3 (3, e cubed). The points (1, 0) and (3, 0) are marked on the x axes. A dashed line rises from (1, 0) marked x = 1; similarly, a solid line rises from (3, 0) marked x = 3.
Finding a centroid of a region below the curve y = e x .

To compute the centroid, we assume that the density function is constant and hence it cancels out:

x c = M y m = R x d A R d A and y c = M x m = R y d A R d A , x c = M y m = R x d A R d A = x = 1 x = 3 y = 0 y = e x x d y d x x = 1 x = 3 y = 0 y = e x d y d x = x = 1 x = 3 x e x d x x = 1 x = 3 e x d x = 2 e 3 e 3 e = 2 e 2 e 2 1 , y c = M x m = R y d A R d A = x = 1 x = 3 y = 0 y = e x y d y d x x = 1 x = 3 y = 0 y = e x d y d x = x = 1 x = 3 e 2 x 2 d x x = 1 x = 3 e x d x = 1 4 e 2 ( e 4 1 ) e ( e 2 1 ) = 1 4 e ( e 2 + 1 ) .

Thus the centroid of the region is

( x c , y c ) = ( 2 e 2 e 2 1 , 1 4 e ( e 2 + 1 ) ) .
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Calculate the centroid of the region between the curves y = x and y = x with uniform density in the interval 0 x 1 .

x c = M y m = 1 / 15 1 / 6 = 2 5 and y c = M x m = 1 / 12 1 / 6 = 1 2

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Moments of inertia

For a clear understanding of how to calculate moments of inertia using double integrals, we need to go back to the general definition in Section 6.6 . The moment of inertia of a particle of mass m about an axis is m r 2 , where r is the distance of the particle from the axis. We can see from [link] that the moment of inertia of the subrectangle R i j about the x -axis is ( y i j * ) 2 ρ ( x i j * , y i j * ) Δ A . Similarly, the moment of inertia of the subrectangle R i j about the y -axis is ( x i j * ) 2 ρ ( x i j * , y i j * ) Δ A . The moment of inertia is related to the rotation of the mass; specifically, it measures the tendency of the mass to resist a change in rotational motion about an axis.

The moment of inertia I x about the x -axis for the region R is the limit of the sum of moments of inertia of the regions R i j about the x -axis . Hence

I x = lim k , l i = 1 k j = 1 l ( y i j * ) 2 m i j = lim k , l i = 1 k j = 1 l ( y i j * ) 2 ρ ( x i j * , y i j * ) Δ A = R y 2 ρ ( x , y ) d A .

Similarly, the moment of inertia I y about the y -axis for R is the limit of the sum of moments of inertia of the regions R i j about the y -axis . Hence

I y = lim k , l i = 1 k j = 1 l ( x i j * ) 2 m i j = lim k , l i = 1 k j = 1 l ( x i j * ) 2 ρ ( x i j * , y i j * ) Δ A = R x 2 ρ ( x , y ) d A .

Sometimes, we need to find the moment of inertia of an object about the origin, which is known as the polar moment of inertia. We denote this by I 0 and obtain it by adding the moments of inertia I x and I y . Hence

I 0 = I x + I y = R ( x 2 + y 2 ) ρ ( x , y ) d A .

All these expressions can be written in polar coordinates by substituting x = r cos θ , y = r sin θ , and d A = r d r d θ . For example, I 0 = R r 2 ρ ( r cos θ , r sin θ ) d A .

Finding moments of inertia for a triangular lamina

Use the triangular region R with vertices ( 0 , 0 ) , ( 2 , 2 ) , and ( 2 , 0 ) and with density ρ ( x , y ) = x y as in previous examples. Find the moments of inertia.

Using the expressions established above for the moments of inertia, we have

I x = R y 2 ρ ( x , y ) d A = x = 0 x = 2 y = 0 y = x x y 3 d y d x = 8 3 , I y = R x 2 ρ ( x , y ) d A = x = 0 x = 2 y = 0 y = x x 3 y d y d x = 16 3 , I 0 = R ( x 2 + y 2 ) ρ ( x , y ) d A = 0 2 0 x ( x 2 + y 2 ) x y d y d x = I x + I y = 8.
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Again use the same region R as above and the density function ρ ( x , y ) = x y . Find the moments of inertia.

I x = x = 0 x = 2 y = 0 y = x y 2 x y d y d x = 64 35 and I y = x = 0 x = 2 y = 0 y = x x 2 x y d y d x = 64 35 . Also, I 0 = x = 0 x = 2 y = 0 y = x ( x 2 + y 2 ) x y d y d x = 128 21 .

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As mentioned earlier, the moment of inertia of a particle of mass m about an axis is m r 2 where r is the distance of the particle from the axis, also known as the radius of gyration    .

Hence the radii of gyration with respect to the x -axis, the y -axis, and the origin are

Questions & Answers

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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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