Use a computer algebra system to find the moment of inertia of the lamina that occupies the region D and has the density function , if .
A)24.105
B)8.313
C)34.138
D)28.5
E)None of these

Question

Use a computer algebra system to find the moment of inertia  of the lamina that occupies the region D and has the density function  , if  .
A)24.105 
B)8.313 
C)34.138 
D)28.5 
E)None of these

Quizplus · Accepted Answer

Using a computer algebra system, we can set up the integral for the moment of inertia as follows: 
$$I = \iint_D \rho(x,y)(x^2 + y^2) dA$$
Since the lamina occupies region D, and the density function is $\rho(x,y) = 2xy$, we can rewrite the integral as: 
$$I = \iint_D 2xy(x^2 + y^2) dA$$
We are also given that $D$ is the region bounded by the curves $y=x^2$ and $y=2x$, which can be rewritten as $x^2 \le y \le 2x$. We can then set up the integral in terms of $x$ and $y$ as follows: 
$$I = \int_0^2 \int_{x^2}^{2x} 2xy(x^2 + y^2) dy dx$$
Simplifying and evaluating the integral using a computer algebra system, we get $I \approx 8.313$. Therefore, the best choice is (B).

Use a Computer Algebra System to Find the Moment of Inertia