Calculate the next limit. Lebesgue integral.

Question

I'm trying to solve the next problem:

Calculate, justifying all steps, the limit $$ \lim_{n \rightarrow \infty} \int_A \dfrac{1+ \dfrac{\cos^2(x^3)}{n} }{x^2+y^2+1} dx \ dy$$ where $A=\{(x,y) \in \mathbb{R}^2 : x^2+y^2 < 4\}$

I tried to changing to polar coordinate. But, I don't know how to follow.

Try this: $$\lim_{n\to +\infty}\left(C+\frac{D}{n}\right) = C = 2\pi\int_{0}^{2}\frac{\rho,d\rho}{\rho^2+1}=\pi\log 5.$$ — Jack D'Aurizio, Jun 08 '16 at 17:15

score 1 · Accepted Answer · answered Jun 08 '16 at 17:17

First, note that as $n \to \infty$, $$ f_n(x,y):=\frac{1+\frac{\cos^2(x^3)}{n}}{x^2+y^2+1} \to \frac{1}{x^2+y^2+1}=:f(x,y) $$ pointwise. Using polar coordinates, \begin{align*} \iint_A \frac{1}{x^2+y^2+1} dA &= \int_0^{2\pi} \int_0^2 \frac{1}{r^2+1} r \, dr \, d\theta \\&= \int_0^{2\pi} \frac{\ln 5}{2} d\theta = \pi \ln 5. \end{align*}

Now why does $\lim_{n \to \infty} \int_A f_n(x,y) dA = \int_A f(x,y) dA$? There are a few ways to show this. For one way, note that $0 \leq f_n(x,y) \leq \frac{2}{x^2+y^2+1}$, which is an integrable function over $A$. Then use Dominated Convergence Theorem to finish it off

Calculate the next limit. Lebesgue integral.

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