43 lines
1.2 KiB
Markdown
43 lines
1.2 KiB
Markdown
# ECE 106: Electricity and Magnetism
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## MATH 117 review
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!!! definition
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A definite integral is composed of:
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- the **upper limit**, $b$,
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- the **lower limit**, $a$,
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- the **integrand**, $f(x)$, and
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- the **differential element**, $dx$.
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$$\int^b_a f(x)\ dx$$
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The original function **cannot be recovered** from the result of a definite integral unless it is known that $f(x)$ is a constant.
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## N-dimensional integrals
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Much like how $dx$ represents an infinitely small line, $dx\cdot dy$ represents an infinitely small rectangle. This means that the surface area of an object can be expressed as:
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$$dS=dx\cdot dy$$
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Therefore, the area of a function can be expressed as:
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$$S=\int^x_0\int^y_0 dy\ dx$$
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where $y$ is usually equal to $f(x)$, changing on each iteration.
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!!! example
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The area of a circle can be expressed as $y=\pm\sqrt{r^2-x^2}$. This can be reduced to $y=2\sqrt{r^2-x^2}$ because of the symmetry of the equation.
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$$
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\begin{align*}
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A&=\int^r_0\int^{\sqrt{r^2-x^2}}_0 dy\ dx \\
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&=\int^r_0\sqrt{r^2-x^2}\ dx
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\end{align*}
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$$
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!!! warning
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Similar to parentheses, the correct integral squiggly must be paired with the correct differential element.
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## Cartesian coordinates
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