Applications of Integration

$\pi\int_{e^{-3}}^{e}{(e^{-3x}) - (e^{-x})} dx$

$\int_0^{\ln(4)}{\pi[(e^{-3x})-(e^{-x})]} dx$

$\pi\int_0^{\ln(4)}{(e^{-x})^2 - (e^{-3x})^2} dx$

$\pi\int_{-3}^{\ln(4)}{(e^{-3/x}/3) - (e^{-1/x}/x)} dx$

$\frac{3\pi}{4}$

$\frac{7\pi}{8}$

$\frac{5\pi}{6}$

$\frac{2\pi}{3}$

$\frac{11\pi}{6}$

$\frac{9\pi}{4}$

$\frac{13\pi}{6}$

$\frac{15\pi}{8}$

$\pi a\left[\frac {\theta}{b}-\sin\left(\frac {\theta}{b}\right)\right]$ evaluated from $0$ to $\frac {\pi}{b}$

$\frac {\pi a}{ b}\left[\theta-\sin(b\theta)\right]$ evaluated from $0$ to $\frac {\pi}{ b}$

$\frac {\pi a^{2}}{ b^{2}}[\eta-\sin(b\eta)]$ evaluated at $\eta=\theta b$

$2\pi\left[\frac {\theta}{b}-\sin\left(\frac {\theta}{b}\right)\right]$ evaluated from $0$ to $\pi b$

$\pi \int_{0}^{4} ((5+x)^2 - (\sqrt{x})^2) dx$

$\pi \int_{0}^{4} (25 - (\sqrt{5-x})^2) dx$

$\pi \int_{0}^{4} ((5-\sqrt{x})^2 - (5-x)^2) dx$

$\pi \int_{0}^{4} (25 - (5-x)^2) dx$

Differentiation of the region's functions

Differentiation of the washer function

Integration of the washer function

Differentiation of the region's width

The idea of a limit as x approaches a value allowing us to approximate area under curves with rectangles

Optimization principles helping determine maximum volumes rather than methods of finding them

The concept of continuity ensuring functions can be graphed without lifting our pencil

The Fundamental Theorem of Calculus linking differentiation and integration directly without consideration of limits

Replace all x terms after solving $y=f^{-1}(X)$ for $X$

Integrate w.r.t. $y$ directly without further substitutions

Add $\pi$ outside Integral sign because it involves circular geometry

Substitute $dy$ for $dx$ after integrating each term

$\pi \int_{0}^{\ln(4)} [(e^{x} +1 )^2 - (xe^{-x}-1)^2]dx$

$\pi \int_{0}^{\ln(4)} [(e^{x})^2 -(xe^{-x})^2]dx$

$\pi \int_{0}^{\ln(4)} [e^{2x +1} - x^2(e^{-x}-1)]dx$

$\pi \int_{-1}^{\ln(3)} [(e^{x+1})^2 -(xe^{-x+1})^2]dx$

Disk Method, as it is suitable for solids with no hole in the middle.

Washer Method, because it accommodates non-constant gap between curves.

Shell Method, since it simplifies computation for rotation around an axis parallel to shapes.

Cross-section Method, as it is often used for simple geometrical shapes.