Area Between Two Curves: Step-by-Step Method and Examples
Area between curves = ∫(top − bottom)dx. Find intersection points, handle curves that cross, and integrate with respect to y when easier. Worked examples.
Tag: integrals
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Riemann Sums: Left, Right, Midpoint and Simpson’s Rule
How to compute left, right and midpoint Riemann sums, the trapezoidal rule and Simpson’s rule, with a full worked example and error comparison.
Average Value of a Function: Formula, Meaning and Examples
The average value of f on [a, b] is (1/(b − a))∫f(x)dx. Why the formula works, examples with sin x, x² and eˣ, and the Mean Value Theorem for integrals.
Shell Method for Volume: Formula, Examples and Disk vs Shell
The shell method finds volumes of revolution with V = 2π∫(radius)(height)dx. Step-by-step examples and a clear guide to choosing shells or disks.
Disk and Washer Method: Volume of Revolution Step by Step
Find volumes of solids of revolution with the disk method V = π∫f(x)²dx and the washer method V = π∫(R² − r²)dx. Worked examples and setup tips.
Arc Length Formula in Calculus: Derivation and Examples
The arc length of y = f(x) from a to b is ∫√(1 + [f'(x)]²) dx. Derivation, worked examples (x^(3/2), a circle, a catenary) and parametric curves.
Fundamental Theorem of Calculus (Parts 1 and 2) Explained
The Fundamental Theorem of Calculus links derivatives and integrals. Both parts explained with intuition, formulas and worked examples.
Improper Integrals: How to Tell If They Converge or Diverge
Improper integrals have infinite limits or infinite integrands. Learn the limit definition, the p-integral test, comparison, and 6 worked examples.
Partial Fraction Decomposition: Steps and Examples
How to integrate rational functions with partial fractions: distinct linear factors, repeated factors and irreducible quadratics, with worked examples.
Integration by Parts: Formula, LIATE Rule and Examples
Integration by parts: ∫u dv = uv − ∫v du. How to choose u with LIATE, 6 worked examples (x·sin x, ln x, arctan x, eˣ·cos x) and the tabular method.