Applications of Integration

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

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

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

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

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.

Substitute $f(x)+[MODIFIER]$ into your radius calculation to account for vertical shift.

Divide overall integral by [MODIFIER] before plugging in values of f(x).

Multiply overall integral by [MODIFIER] after evaluating it from bounds [$a,b$].

Add [MODIFIER] directly inside your integral bounds changing them from [$a+[MODIFIER],b+[MODIFIER]$].

Sum Law of Limits

Limit Comparison Test

Squeeze theorem

Intermediate Value theorem

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

$\frac{2\pi}{5}$

$\frac{5\pi}{2}$

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

By subtracting the smaller disk's volume from the larger disk's volume

By summing up the areas of the two bases and the lateral area

By multiplying the width by the average radius

By integrating the function for the volume of each washer

$\pi \int_{-5}^{-5} ((3-y)-(y-1))^{[MISSING EXPONENT]} dy$

$\pi \int_1^3 ((3-y)^{2}-(y-1)^{2}) dy$

$\pi \int_{0}^{4} ((y-3)-(1+y))^{[MISSING EXPONENT]} dy$

$\pi \int_1^{3} ((3-x)^{2}-(1+x)^{2}) dx$

Integrate with respect to $x$

Differentiate with respect to $y$

Differentiate with respect to $x$

Integrate with respect to $y$

They must alternately be increasing and decreasing over [a, b]

They can be discontinuous but still increasing over [a, b]

They must be continuous and non-negative over [a, b]

They should intersect at some point within [a, b]

The Washer Method

The Section Method

The Revolution Method

The Slicing Method