List of formulas that may be useful for your queries:
- Absolute convergence of an improper integral:
[
\int_a^\infty |f(x)| \, dx < \infty \implies \int_a^\infty f(x) \, dx \text{ converges absolutely.}
] - Conditional convergence of an improper integral:
[
\int_a^\infty f(x) \, dx \text{ converges, but } \int_a^\infty |f(x)| \, dx = \infty.
] - Improper integral in polar coordinates:
[
\int\int_D f(r, \theta) \, r \, dr \, d\theta,
]
where ( D ) is the domain in polar coordinates. - Parametric representation of a curve:
If a curve is given parametrically as ( \mathbf{r}(t) = (x(t), y(t)) ), then the arc length ( S ) from ( t=a ) to ( t=b ) is calculated by the formula:
[
L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt.
] - Arc length of a curve in polar coordinates:
If a curve is given in polar coordinates as ( r = f(\theta) ), then the arc length from ( \theta = a ) to ( \theta = b ) is calculated by the formula:
[
L = \int_a^b \sqrt{f(\theta)^2 + \left(\frac{df}{d\theta}\right)^2} \, d\theta.
] - Surface area of revolution:
If the curve is given parametrically, then the surface area of revolution about the axis ( x ) is calculated by the formula:
[
S = 2\pi \int_a^by(t) \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt.
]