1.2 KiB
ECE 106: Electricity and Magnetism
MATH 117 review
!!! definition A definite integral is composed of:
- the **upper limit**, $b$,
- the **lower limit**, $a$,
- the **integrand**, $f(x)$, and
- the **differential element**, $dx$.\[\int^b_a f(x)\ dx\]
The original function cannot be recovered from the result of a definite integral unless it is known that \(f(x)\) is a constant.
N-dimensional integrals
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:
\[dS=dx\cdot dy\]
Therefore, the area of a function can be expressed as:
\[S=\int^x_0\int^y_0 dy\ dx\]
where \(y\) is usually equal to \(f(x)\), changing on each iteration.
!!! example 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.
$$
\begin{align*}
A&=\int^r_0\int^{\sqrt{r^2-x^2}}_0 dy\ dx \\
&=\int^r_0\sqrt{r^2-x^2}\ dx
\end{align*}
$$!!! warning Similar to parentheses, the correct integral squiggly must be paired with the correct differential element.