Fundamental Theorems of Integral Calculus and their Applications

Derive the formula for the volume of a cylinder with radius $\text{'}r\text{'}$ and height $\text{'}h\text{'}$ by using integration.

Prove that ${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\frac{dx}{1+\sqrt{\cot x}}={\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\frac{dx}{1+\sqrt{\tan x}}$.

Evaluate $\int {\cos }^{5}xdx$

A cup of tea at temperature $100°C$ is placed in a room whose temperature is $15°C$ and it cools to $60°C$ in $5$ minutes. Find its temperature after further interval of $5$ minutes.

Prove that $F\left(3\right)=1-e^{-9}$ if the probability density function $f\left(x\right)$ is defined as $f\left(x\right)=\left\{\begin{array}{l}3e^{-3x}, x>0\\ 0, x\le 0\end{array}\right.$ ,

The value of ${\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{\tan x-\cot x}{1+\tan x \cot x}dx$ is:

The value of ${\int }_0^{\pi }{\sin }^{4}x dx$ is