Use $R_4$ to estimate the area under the curve $y= \frac{2}{1+x^2}$ between $x=0$ and $x=1$.

Question

Question : Use $R_4$ to estimate the area under the curve $y= \frac{2}{1+x^2}$ between $x=0$ and $x=1$.

Not sure how to proceed with this question.

Answer 1

Let $$f(x)=\frac2{(1+x^2)}$$ Between $0$ and $1$ we have the width of each section equal to $\frac14$ because we are using $4$ subsections.

For the first subsection we have width times height so $\frac14\times f(0.25)$
($0.25$ because we are using $R$-approximations).

We increment the input for the function by $0.25$ until we reach $1$, and then we sum up all the areas. Observe $$[0.25 \times f(0.25)] + [0.25 \times f(0.5) ]+ [0.25 \times f(0.75)] + [0.25 \times f(1)]$$

Factor the $0.25$ out we have
$= 0.25(f(0.25) + f(0.5) + f(0.75) + f(1))$

$= 0.25(\frac{32}{17} + \frac85 + \frac{32}{25} + 1)$

$= \frac{2449}{1700}$

$\approx 1.44$

Diagram