PROBLEMS: FASTΒΆ

<td>$\displaystyle \int \ln(x)dx = x\ln(x) - x + C$</td>

<td>$\displaystyle \int \sin(x) dx = -\cos(x) + C$</td>
<td>$\displaystyle \int \cos(x) dx = \sin(x) + C$</td>
<td>$\displaystyle \int a^x dx = \frac{a^x}{\ln a} + C$</td>
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CONTINUOUS PROBLEMS

  1. Area under \(f(x) = x^5\) from \(x = 0\) to \(x = 1\).
  2. Area under \(g(x) = \sin(x) + 4\) from \(x = \pi\) to \(x = 2\pi\)
  3. Area between \(f(x) = x\) and \(g(x) = \sqrt{x}\)
  4. Volume formed by rotating region in 3 about the \(x\)-axis.
DISCRETE PROBLEMS 1. How many ways are there to throw 4 heads when tossing 6 coins? Explain.
2. Describe in your own words what the following two equations mean and what we use them for:
\[\mu = \frac{1}{n} \sum_{i = 1}^n x_i \quad \sigma^2 = \frac{1}{n}\sum_{i = 1}^n (x_i - \mu)^2\]

Do \(\mu\) and \(\sigma^2\) have other names?

  1. A tank has a height of 10 feet. The area of a horizontal cross section of the tank at height \(h\) is given by the function \(A\) where \(A(h)\) is measured in square feet. Use a riemann sum to approximate the volume of the tank.

\(\displaystyle \int x^n dx = \frac{x^{n+1}}{n+1} + C\)

\(\displaystyle \int e^{ax} dx = \frac{1}{a} e^{ax} + C\)
h A(h)
0 50.3
2 13.2
5 5.5
10 3.0

EXTRA

  • CODE: Answer any of the problems using python code.
  • DATA: Is America getting more or less equal? Answer this question using census data and riemann summations to approximate gini indicies.
  • DESIGN: Using 3D functions, you should design a structure on specified domains. Use Python and our earlier examples to build a slider that allows structure to be viewed from multiple angles.