Extra credit on integration

This is worth 5 points extra credit if done properly and turned in by May 18.

Get this as an ipynb file for IJulia.

Definite integration can be done different ways:

$$~ \int_a^b f(x) dx = F(b) - F(a) ~$$

In Julia, the SymPy package can find antiderivatives and evaluate, but that isn't needed here.

If a numeric answer is sought, then the quadgk function can provide an answer using an algorithm more sophisticated than a Riemann sum. This function gives two answers: the answer and an estimate on the maximum possible error between the true answer and the numerical estimate.

Suppose $f(x) \geq 0$. Then different definite integrals have different interpretations. The Riemann sum approach uses the fact that

The definite integral $\int_a^b f(x) dx$ is the area under $f(x)$ over the interval $[a,b]$.

Some integrals give different values. For example

The definite integral $\int_a^b \sqrt{1 + f'(x)^2} dx$ give the arc length of the graph of $f(x)$ between $a$ and $b$.

Questions

You can use the MTH229 package.

using MTH229

Use the quadgk function (quadgk(f, a, b) or quadgk(f,a,b)[1] to get just the answer) to show that the arc length of the function $f(x) = 2 + 3x$ over $[0,4]$ is the same as given by the distance formula between the points $(0, f(0))$ and $(4, f(4))$.

The function cosh implements the hyperbolic cosine. Make a plot of the graph over the interval $[-2, 3]$. Estimate the area under the graph. (Is it about 6, 10, 14, 20, ...?)

From the same graph estimate the arc length of the graph. (It is more than 5 and less than 19 = 4 + 5 + 10.)

Use quadgk to numerically find the area under the cosh function over $[-2, 3]$.

Use quadgk to numerically find the arc length of the graph of the cosh function over $[-2, 3]$.