Evaluate the following summation (make sure your calculator is in radian mode): 
${\sum }_{i=1}^4\sin \left(\frac{\pi i}{8}\right)$

Heat capacity of a gas

Heat capacity 

C_v

is the amount of heat required to raise the temperature of a given mass of gas with constant volume by 1°C, measured in units of cal/deg-mol (calories per degree gram molecular weight).

The heat capacity of oxygen depends on its temperature T and satisfies the formula

C_v = 8.27 + 10^(-5) * (26T − 1.87T²)

Use Simpson’s Rule to find the average value of C_v and the temperature at which it is attained for

20°C ≤ T ≤ 675°C.

Evaluate the following summation: 

${\sum }_{i=1}^4{i}^2-3i+8$

Evaluate the following summation: 

${\sum }_{k=1}^4{\left(\frac{k}{{}_{}2}\right)}^2$

Evaluate the following summation: 

${\sum }_{i=0}^2\frac{2i}{3}$

Evaluate the following summation: 

${\sum }_{t=2}^{6}5t+\frac{4}{t}$

Evaluate the following summation: 

${\sum }_{i=0}^{4}2i+1$

Evaluate the following summation: 

${\sum }_{i=1}^{2}4i^3-3i^2+2i-1$

Evaluate the following summation (make sure your solution is in radians):

${\sum }_{k=0}^{5}2\tan \left(\frac{\pi k}{3}\right)-\pi k$