Applications of Integration

Disk Method, since we are rotating around a perpendicular axis to variable bounds.

Cross-sections Method, as it's not specific to rotation problems.

Washer Method, as it deals with solids with holes in them.

Shell Method, since it involves integrating with respect to y.

By taking the square root of the radius of the disc.

By multiplying the area of the disc by the width of the disc.

By approximating the volume using Riemann sums.

By finding the derivative of the function.

Shell method would account for cylindrical shells instead of discs leading to a different integral setup but same final result after solving.

Shell method underestimates because it approximates smaller sections ignoring large gaps between shells at wider parts of hyperboloid.

Disc method results in an overestimation compared to shell due to adding more material at larger radii where hyperboloid tapers off.

Neither affects calculated volume significantly as both methods approach actual value closely enough for practical purposes in calculus level precision requirements.

A cone

A sphere

A disk

A cylinder

Add both curves together prior to squaring regardless of their relative positions.

Always use absolute value differences irrespective rotation direction or relative position

Subtract lower curve from upper curve before squaring if rotating about horizontal axis.

Subtract higher curve from lower curve before squaring if rotating about horizontal axis.

(4π)

(12π)

(16π/3)

(8π)

Calculating the area under a curve.

Finding the volume of a solid formed by revolving a region around an axis.

Evaluating limits of functions.

Determining the slope of a tangent line.

Computing the definite integral of $\sqrt{x}$ without any alterations.

Differentiating $\sqrt{x}$ before squaring it in order to find velocity.

Squaring the function $\sqrt{x}$ to obtain $(\sqrt{x})^2 = x$ as part of integrand.

Integrating $\frac{1}{\sqrt{x}}$ across bounds as it's reciprocal function.

Implicitly rotating axes themselves until parallel with coordinate axes, then applying method normally.

Using directly linear expressions for each component as in surface area calculations.

Including both radial and curvilinear components as in cross-sectional integration.

Adjusting radius expression to account for perpendicular distance from rotation axis.

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

$\frac{1}{3}\pi\left[x(\sqrt{x})^3\right]_0^4$

$\pi \int_{0}^{4}\sqrt{x}dx$

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