A lamina occupies the part of the disk $x^2+y^2\leq 25$ in the first quadrant.

Question

Here is the question: A lamina occupies the part of the disk $x^2+y^2\leq 25$ in the first quadrant. Its density is $\rho=4$.

1. What is the total mass?
2. What is the moment about the x-axis?
3. What is the moment about the y-axis?

Thanks a ton.

Answer 1

HINT

For the first question, consider the integral

\begin{align*} M = \iint_{R}\rho(x,y)\mathrm{d}y\mathrm{d}x = 4\int_{0}^{5}\int_{0}^{\sqrt{25-x^{2}}}1\mathrm{d}y\mathrm{d}x \end{align*}

For the second question, consider \begin{align*} I_{x} = \iint_{R}r^{2}\mathrm{d}m = \iint_{R}y^{2}\rho(x,y)\mathrm{d}y\mathrm{d}x = 4\int_{0}^{5}\int_{0}^{\sqrt{25-x^{2}}}y^{2}\mathrm{d}y\mathrm{d}x \end{align*}

Finally, for the last question, we have \begin{align*} I_{y} = \iint_{R}r^{2}\mathrm{d}m = \iint_{R}x^{2}\rho(x,y)\mathrm{d}y\mathrm{d}x = 4\int_{0}^{5}\int_{0}^{\sqrt{25-x^{2}}}x^{2}\mathrm{d}y\mathrm{d}x \end{align*}