Question

I think I did the first two right but need help with the length of the curve.

Answer 1

We have the following function, f(x)

$\sqrt{1+\left(\frac{1}{\sqrt{1-x^2}}\right)^2}$

and, we need to find the riemann sum with n=3

$\int _{-\frac{1}{4}}^{\frac{1}{4}}\sqrt{1+\left(\frac{1}{\sqrt{1-x^2}}\right)^2}dx$

Let's use the following to find the right riemann sum

$\int_a^bf\left(x\right)dx\:\approx\sum_{n\mathop{=}1}^3f(x_i)*\Delta x$

1st, let's calculate dx

$\Delta x=\frac{b-a}{n}=\frac{0.25+0.25}{3}=\frac{1}{6}$

2nd, calculate each f(xi)

$\begin{gathered} x_1=-\frac{1}{4}+\frac{1}{6}=-\frac{1}{12},f(x_1)=\sqrt{\frac{287}{143}} \\ x_2=-\frac{1}{12}+\frac{1}{6}=\frac{1}{12},f(x_2)=\sqrt{\frac{287}{143}} \\ x_3=\frac{1}{12}+\frac{1}{6}=\frac{1}{4},f(x_3)=\sqrt{\frac{31}{15}} \end{gathered}$

Now, let calculate the right riemann sum

$\sum_{n\mathop{=}1}^3f(x_i)*\Delta x=\frac{1}{6}*\left(\sqrt{\frac{287}{143}}+\sqrt{\frac{287}{143}}+\sqrt{\frac{31}{15}}\right)$

Solving, we get

$=0.7118$

Thus, the answer is 0.7118