Applications of Integration

Area under curve integral

Arc length integral

Surface area integral

Volume rotational integral

$r = x + y$

$y = r + x$

$r = |x|$

$x = y - r$

The speed function over time

The acceleration function over time

The velocity function squared over time

The original position functions over time

Improper integral.

Integral involving partial fractions.

Definite integral.

Indefinite integral.

$\int_{}^{} \sqrt{(\sinh(t))^2 + (\cosh(t))^2} dt$

\int_{}^{} \sqrt{(\sinh(\log)) + \cos(n)p(t)) dt

$\int_{}^{} \cosh(t) \tan(dt) dT$

$\int_{}^{} \sqrt{(\cosh(\tan(t))^2 + (\sinh(t))^2} dt$

Sum up all infinitesimal distances $(dx)^{-e^{(-x)}}$ from $x=0$ to infinity

Calculate $\frac{1}{e^{-x}}$ evaluated from $x=0$ to $+\infty$

Evaluate $\lim_{a \to +\infty} \int_0^{a} e^{-x} dx$

Evaluate $\lim_{a \to +\infty} \int_0^{a} \sqrt{1+e^{-2x}} dx$

The curve must be linear

The curve must be planar

The curve must be smooth

The curve must be continuous

There must exist two points in [c,d] where g(t) has equal values.

$g(t)$ must have an inverse on [c,d].

$g(t)$ must also be differentiable on [c,d].

$g(t)$'s range must be all real numbers.

Leverage Cauchy's Integral Formula derivations tailored towards real-valued arc length interpretations from complex paths.

Use brute force evaluation circumventing singularity constraints inherent in closed-path scenarios applicable here.

Apply residue theorem indiscriminately disregarding contour restrictions relevant only within real plane analyses.

Introduce branch cuts arbitrarily aimed at reducing multi-valued function issues neglecting path connectivity.

$\int_{\pi}^{2\pi} \sqrt{\left(e^\theta\right)^{2}+{\left(\frac{d}{d\theta}(e^\theta)\right)}^{2}} d\theta$

\int_{e^\pi}^{e^{(▶️)}}▷️ \◁️ \₂π} !!! ▢️ (θ)dθ

\int_{{3π}. π⇾↩⬅️⟿⟾↪↔⇿₄∑ ₓ₀²+({ⅆᵣ)/ⅆθ})↑₂₁₀τdτ

$\int_{\pi}^{e^\pi} \sqrt{\left(e^\theta\right)^{2}} d\theta$