CalculusLesson 1 of 2 in the course Integrals

Integral Concepts, Properties, Methods, and Applications

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

A finite sum adds separate values. An integral extends this idea to a continuum: a region or interval is divided into many small pieces, the contribution of each piece is calculated, and the pieces are added. As the pieces become narrower, the sum can approach an exact value.

For example, to estimate the area under f(x)=xf(x)=\sqrt{x} from x=0x=0 to x=1x=1, divide the interval into narrow sections and use each section's width times a representative function value. Taking the limit as the section widths approach zero gives ∫01x dx=23\int_0^1 \sqrt{x}\,dx=\frac{2}{3}.

All flashcards

What is an integral, intuitively?
A continuous analogue of a sum that accumulates small contributions to calculate quantities such as area or displacement.
What is a definite integral?
An integral with specified bounds that measures the signed accumulation of a function across an interval.
What is an indefinite integral?
A family of antiderivatives of a function, written without bounds and including an arbitrary constant.
What does dxdx indicate in an integral?
Integration is performed with respect to xx.
What is a Riemann sum?
A sum of sampled function values multiplied by subinterval widths, used to approximate an integral.

Quiz questions

  1. If F′(x)=f(x)F'(x)=f(x) on [a,b][a,b], how can the fundamental theorem of calculus evaluate ∫abf(x) dx\int_a^b f(x)\,dx?Answer: F(b)−F(a)F(b)-F(a)
  2. For an integrable function, what happens when the bounds of a definite integral are reversed?Answer: The value changes sign.
  3. How does the Lebesgue approach to integration differ from the Riemann approach, in broad terms?Answer: It groups points according to function values and uses measure.
  4. When does an improper integral converge?Answer: When the defining limit exists and is finite.
  5. Under suitable conditions, what does Fubini's theorem allow for a double integral over a rectangle?Answer: It can be evaluated as an iterated integral.

Based on the Wikipedia article Integral, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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