(a) Evaluate the integral
$\displaystyle {\int_{{{0}}}^{{{2}}}}{\frac{{{40}}}{{{x}^{{2}}+{4}}}}{\left.{d}{x}\right.}$.
Your answer should be in the form $\displaystyle {k}\pi$, where $\displaystyle {k}$ is an integer. What is the value of $\displaystyle {k}$?
Hint: $\displaystyle {\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\arctan{{\left({x}\right)}}}={\frac{{{1}}}{{{x}^{{2}}+{1}}}}$
$\displaystyle {k}=$ Preview Question 6 Part 1 of 8  


(b) Now, lets evaluate the same integral using power series. First, find the power series for the function $\displaystyle {f{{\left({x}\right)}}}={\frac{{{40}}}{{{x}^{{2}}+{4}}}}$. Then, integrate it from 0 to 2, and call it S. S should be an infinite series.
What are the first few terms of S ?
$\displaystyle {a}_{{0}}=$ Preview Question 6 Part 2 of 8  
$\displaystyle {a}_{{1}}=$ Preview Question 6 Part 3 of 8  
$\displaystyle {a}_{{2}}=$ Preview Question 6 Part 4 of 8  
$\displaystyle {a}_{{3}}=$ Preview Question 6 Part 5 of 8  
$\displaystyle {a}_{{4}}=$ Preview Question 6 Part 6 of 8  


(c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by $\displaystyle {k}$ (the answer to (a)), you have found an estimate for the value of $\displaystyle \pi$ in terms of an infinite series. Approximate the value of $\displaystyle \pi$ by the first 5 terms.
Preview Question 6 Part 7 of 8   .


(d) What is the upper bound for your error of your estimate if you use the first 9 terms? (Use the alternating series estimation.)
Preview Question 6 Part 8 of 8   .