Applications of Integration

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

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

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

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

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

$\pi \int_0^2 (x^4) - (4x)^2 dx$

$\pi \int_0^2 (2x)^2 - (x^2)^2 dx$

$\pi \int_0^4 (x^2) - (2x)^2 dx$

$\pi \int_0^4 (x^4) - (4x) dx$

6π

4π

10π

8π

(2π)

(3π)

(5π)

(4π)

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.

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

Washer method factoring shift leftwards adds constant inside integral expression accounting for new axis location.

Approximation using geometric figures like prisms or cylinders without considering exact curvature provided by graphs' equations.

Disk method assuming original coordinate axes thus ignoring horizontal translation required here.

Shell method preferring cylindrical surfaces aligned along rotation line but overlooks consistent gap between 'shells'.

$\frac{16\pi}{3}$

$\frac{8\pi}{3}$

$\frac{32\pi}{3}$

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