The volume V of a rectangular prism with length l, width w, and height h is v equals l times w times h. What is the width of a r
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The moment of inertia about the y-axis of the thin semicircular region of constant density is given below.
Any item that can be turned has rotational inertia as a quality. It's a scalar value that indicates how complex it is to adjust an object's rotational velocity around a certain axis.
Then the moment of inertia about the y-axis of the thin semicircular region of constant density will be
x = r cos θ
y = r sin θ
dA = r dr dθ
Then the moment of inertia about the x-axis will be
On integration, we have
Then the moment of inertia about the y-axis will be
On integration, we have
Then the moment of inertia about O will be
More about the rotational inertia link is given below.
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First, recall that Gaussian quadrature is based around integrating a function over the interval [-1,1], so transform the function argument accordingly to change the integral over [1,5] to an equivalent one over [-1,1].
So,
Let
. With 
, we're looking for coefficients 
and nodes 
, with 
, such that
You can either try solving for each with the help of a calculator, or look up the values of the weights and nodes (they're extensively tabulated, and I'll include a link to one such reference).
Using the quadrature, we then have
So,
Let
You can either try solving for each with the help of a calculator, or look up the values of the weights and nodes (they're extensively tabulated, and I'll include a link to one such reference).
Using the quadrature, we then have