Integration

Answer all the questions below and press submit to see how many you got right.

id: 18984

$$\int\limits_{0}^{16} x dx=\lim_\limits{n\to\infty} \frac{16}{n}\sum\limits_{k=1}^{n}\frac{16\cdot k}{n}=$$ $$=\frac{16\cdot k}{n}\lim_\limits{n\to\infty} \frac{16^2}{n^2}\sum\limits_{k=1}^{n} k=$$ $$=\lim_\limits{n\to\infty} \frac{16^2}{n^2}\frac{n\cdot(n+1)}{2}=?$$

id: 18901

Which of the following are true for two functions $f$ and $g?$

  • $\int\limits_{0}^{5}f(x)-g(x)dx=\int\limits_{0}^{5}f(x)dx-\int\limits_{0}^{5}g(x)dx$
    $\int\limits_{0}^{5}f(x)\cdot g(x)dx=\int\limits_{0}^{5}f(x)dx \cdot \int\limits_{0}^{5}g(x)dx$
    $\int\limits_{0}^{5}\frac{f(x)}{g(x)}dx=\frac{\int\limits_{0}^{5}f(x)dx}{\int\limits_{0}^{5}g(x)dx}$
    $\int\limits_{0}^{5}f(x)+g(x)dx=\int\limits_{0}^{5}f(x)dx+\int\limits_{0}^{5}g(x)dx$
id: 18881

What is $\lim_\limits{n\to\infty} \frac{5}{n}\sum\limits_{k=1}^{n}f(\frac{k\cdot 5}{n})?$

  • $5\int\limits_{0}^{5}f(x)dx$
    $\int\limits_{0}^{5}f(x)dx$
    5
    0