# MAT 142 Day 1

## 1584 days ago by kcrisman

Welcome to MAT 142!  Let's think about $$\int_0^1\sqrt{1+4x^2}dx$$ and what it might mean for us.

First, can we numerically approximate it?  Here is a left-hand sum with $m=20$ subintervals.

m=20 approx=sum([sqrt(1+4*((i-1)^2)/m^2)*1/m for i in [1..m]]) show(approx); n(approx)
 \newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{20} \, \sqrt{2} + \frac{1}{40} \, \sqrt{5} + \frac{1}{40} \, \sqrt{13} + \frac{1}{100} \, \sqrt{26} + \frac{1}{100} \, \sqrt{29} + \frac{1}{100} \, \sqrt{34} + \frac{1}{100} \, \sqrt{41} + \frac{1}{100} \, \sqrt{61} + \frac{1}{100} \, \sqrt{74} + \frac{1}{100} \, \sqrt{89} + \frac{1}{200} \, \sqrt{101} + \frac{1}{100} \, \sqrt{106} + \frac{1}{200} \, \sqrt{109} + \frac{1}{200} \, \sqrt{149} + \frac{1}{200} \, \sqrt{181} + \frac{1}{200} \, \sqrt{221} + \frac{1}{200} \, \sqrt{269} + \frac{1}{200} \, \sqrt{389} + \frac{1}{200} \, \sqrt{461} + \frac{1}{20} 1.44841384355230 1.44841384355230

What exactly does this represent?

P = plot(sqrt(1+4*x^2),(x,0,1),fill=True,figsize=[3,3]) for i in [1..m]: P += line([((i-1)/m,0),((i-1)/m,sqrt(1+4*((i-1)^2)/m^2)),((i)/m,sqrt(1+4*((i-1)^2)/m^2)),(i/m,0)],color='red') show(P)

There are tools to have arbitrary approximation of such an integral, of course.

numerical_approx(integral(sqrt(1+4*x^2),x,0,1))
 1.47894285754460 1.47894285754460

Or we can try for a symbolic integration, and check what that approximates to.

show(integral(sqrt(1+4*x^2),x,0,1)); n(integral(sqrt(1+4*x^2),x,0,1))
 \newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, \sqrt{5} + \frac{1}{4} \, {\rm arcsinh}\left(2\right) 1.47894285754460 1.47894285754460

Well, it turns out this is actually the length of a curve!

plot(x^2,(x,0,1))+plot(x,(x,0,1),linestyle='--',color='red')

 (1.478942857544597, 1.6419564125509345e-14) (1.478942857544597, 1.6419564125509345e-14)
 \newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, \sqrt{4 \, x^{2} + 1} x + \frac{1}{4} \, \log\left(2 \, x + \sqrt{4 \, x^{2} + 1}\right) \newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, \sqrt{5} + \frac{1}{4} \, \log\left(\sqrt{5} + 2\right) 1.47894285754460 1.47894285754460