Applications of Integration

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x

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To determine if the function describing the solid has any points of discontinuity.

To optimize the radius of each washer to maximize material usage.

To calculate the exact volume by approximating with increasingly smaller cylindrical washers.

To ensure that each washer has equal thickness for simpler integration.

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.

$dy$

$\int$

$\pi$

$dx$

It calculates individual washer volumes without considering their relationship along an axis.

Limits only determine rotational symmetry necessary for washer method applicability.

A limit helps define the accumulation points as we add up volumes of infinitesimal washers from start to end point along an axis.

It sets fixed upper and lower bounds on washer sizes within a finite range.

$\pi \int_{0}^{1} [(\sqrt{x})^{3}-x]dx$

$\pi \int_{0}^{1} [(1-\sqrt{x})^{2}-(1-x)^{2}]dx$

$\pi \int_{0}^{1} [(\sqrt{x})^{2}-x^{2}]dx$

$\pi \int_{0}^{1} (\sqrt{x}-x)dx$

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

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

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

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