Applications of Integration

$\pi \int_{0}^{3} \left( \frac{x^4}{3} \right)^2 - \left( \frac{4}{x} \right)^2 dx$

$\pi \int_{0}^{3} \left( \frac{x^4}{3} \right)^3 - \left( \frac{4}{x} \right)^2 dx$

$\pi \int_{0}^{3} \left( \frac{x^4}{3} \right) - \left( \frac{4}{x} \right)^2 dx$

$\pi \int_{0}^{3} \left( \frac{2}{x} - x^2 \right) dx$

$V = \pi \int_{a}^{b} [R(x) + r(x)]^2 dx$

$V = 2\pi \int_{a}^{b}[R(x) - r(x)]dx$

$V = \frac{1}{3} \pi \int_{a}^{b} [R(x)]^2 - [r(x)]^2 dx$

$V = \pi \int_{a}^{b} [R(x)]^2 - [r(x)]^2 dx$

The sum of the volumes of all the washers.

The sum of the radii of all the washers.

The sum of the heights of all the washers.

The sum of the widths of all the washers.

They signify constant radii dimensions within which washers fit perfectly without further calculations required by limits.

Limits at vertical asymptotes determine maximum height but do not influence volumetric calculations otherwise.

Asymptotes indicate where functions become undefined, affecting volume calculation through improper integrals needing limit consideration.

Vertical asymptotes always represent boundaries between solids being integrated separately.

$\frac{16\pi}{5}$

$\frac{8\pi}{5}$

$\frac{7\pi}{2}$

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

The circumference of the washer.

The difference between the inner and outer radii.

The difference between the heights of the washer.

The sum of the inner and outer radii.

y

t

x

z

$\frac{\pi}{4} e^2$

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

$\frac{\pi}{2} e^2$

$2\pi e^2$

$dy$

$\int$

$\pi$

$dx$