Numerical Integration Assignment.sagews

Numerical Integration Assignment

Question 0

Watch the lecture video here.

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Question 1

The Golden Gate Bridge has a main span of 4,200 feet (the distance between the two towers). The main suspension cables that support the road over this span each form a parabolic shape. The length of each cable is found by $$\mathrm{Length}=\int_0^{21}200\sqrt{1+0.000577x^2}\, dx$$

Part a

Approximate the value of this integral using left and right Riemann sums, the Midpoint Rule, the Trapezoidal Rule, and Simpson's Rule using $n=10$, $n=20$, and $n=30$.

Part b

The actual value of this integral is approximately $4371.87$. Observe how close each approximation comes to the right answer. In particular, notice how accurate Simpson's Rule proves to be, even though these values of $n$ are relatively small.

Question 2

Consider the integral $$\int_{-1}^{1.41}\frac{1}{x^2}\, dx$$

Part a

Use the Midpoint Rule with $n=10$, $n=50$, $n=100$, and $n=500$ to approximate this integral.

Part b

You should see your answers jump around as $n$ increases (watch out for scientific notation when $n=100$).

The function $\frac{1}{x^2}$ is unbounded at $x=0$ (we'll learn more about so-called "improper integrals" later this semester). This integral is infinite, so approximation won't work.

Question 3

Consider the function $\displaystyle f(x)=x^5e^{-2x}$ over the interval $[-2,2]$.

[Caution: Don't forget parentheses: e^(-2*x)]

Part a

Approximate the area under this curve using the Midpoint Rule, Trapezoidal Rule, and Simpson's Rule using $n=100$, $n=200$, and $n=400$.

Part b

Rounding to one decimal place, this area is actually $-363.2$. Notice that Simpon's Rule is correct for all three values of $n$, but the Midpoint Rule is correct only for $n=400$, and the Trapezoidal Rule is not correct for any of these values of $n$.

Question 4

Approximate $\displaystyle\int_{-1000}^{1000}\frac{1}{x^2+1}\,dx$ using Simpson's Rule. Hint: The interval width is 2,000, so pick an appropriate number of subintervals ($n$ needs to be several thousand).

What well-known number is this close to?

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is
licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.