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Find the coordinates of the center of mass of the $\frac{1}{4}$ disc defined by

$$\{(x, y) | x\ge 0, y\ge 0, x^2 + y^2 \le 1\}$$

assuming the density is uniform.


Compute $$I=\int \frac{x\cos{x}-\sin{x}}{x^2 + \sin^2{x}} dx$$


Evaluate $$I=\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{1}{\tan\theta +\cot\theta}d\theta$$

Evaluate $$\int_{0}^{\pi}\frac{x\sin{x}}{1+\cos^2 x}dx$$


Let $f(x)=\int_1^x\frac{\ln{x}}{1+x}dx$ for $x > 0$. Find $f(x)+f(\frac{1}{x})$.


Let $f(x)=\int_1^x\frac{\ln{x}}{1+x}dx$ for $x > 0$. Find $f(2)+f(\frac{1}{2})$.


Compute

$$\int_0^{\infty}\frac{x^2}{1+x^4}dx$$


Evaluate $\displaystyle\lim_{n\to\infty}S_n$ where

$$S_n = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots + (-1)^{n-1}\frac{1}{n}$$


Evaluate

$$\int_0^1 x\arcsin{x}d{x}$$


Evaluate

$$\int_0^1 \sqrt{1-x^2} d{x}$$


Compute

$$I= \iiint \limits_S \frac{dx dy dz}{(1+x+y+z)^2}$$

where $S=\{x\ge 0, y\ge 0, z\ge 0, x+y+z\le 1\}$.


Compute $$\int \ln{x} dx$$


Which one of the numbers below is larger?

$$\int_0^{\pi} e^{\sin^2x}dx\qquad\text{and}\qquad \frac{3\pi}{2}$$


Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a periodic continuous function of period $T > 0$, that is $f(x+T)=f(x)$ holds for any $x\in\mathbb{R}$. Show that

$$\lim_{x\to\infty}\frac{1}{x}\int_0^xf(t)dt=\frac{1}{T}\int_0^Tf(t)dt$$


For what pairs $(a, b)$ of positive real numbers does the the following improper integral converge?

$$\int_b^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)dx$$


For $n=1, 2,\dots$, let $x_n=\displaystyle\sum_{k=n+1}^{9n}\frac{k}{9n^2 + k^2}$. Find the value of $\displaystyle\lim_{n\to\infty}x_n$.


A right circular cone $\mathbb{C}$ has altitude $40$ and a circular base of radius $30$ inches. A sphere $\mathbb{S}$ is inscribed in $\mathbb{C}$. Compute the volume of the region inside $\mathbb{C}$ which is above $\mathbb{S}$.


Let $$S_n=\sum_{k=1}^{2n}\frac{1}{n+k}=\frac{1}{n+1}+\frac{1}{n+2}+\cdots + \frac{1}{3n}$$

Does $\displaystyle\lim_{n\to\infty}S_n$ exist? If so, find its value. If not, prove the claim.


According to Newton’s law of cooling, the rate at which a cup of coffee cools is proportional to the difference between its temperature and that of the room it is in. A certain cup of coffee cools from $164^{\circ}$ to $140^{\circ}$ (all temperatures Fahrenheit) in five minutes, and then from $140^{\circ}$ to $122^{\circ}$ in the next five minutes. What is the temperature of the room?


Show the following result without explicitly performing the integration: $$\lim_{n\to\infty}\int_0^1(1-x^2)^ndx = 0$$


Show the following result without explicitly performing the integration $$\lim_{n\to\infty}\int_0^{\frac{\pi}{2}}\sin^n{x}dx$$


Compute $$\int_0^{\frac{\pi}{4}}(\cos{x} - 2\sin{x}\sin(2x))dx$$


Turn the graph of $y=\frac{1}{x}$ by $45^{\circ}$ counter-clockwise and consider the bowl-like top part of the curve (the part above $y=0$). We let a $2D$ fluid accumulate in this $2D$ bowl until the maximum depth of the fluid is $\frac{2\sqrt{2}}{3}$. What’s the area of the fluid used?


Calculate $$\lim_{n\to\infty}\frac{1}{n^2}\sum_{k=1}^{n}\left(k\sin\frac{k\pi}{n}\right)$$


Compute $$\int_0^4\frac{dx}{\sqrt{|x-2|}}$$