CalculusLesson 1 of 1 in the course Multivariable calculus

Multiple Integrals: Iterated Methods, Coordinate Changes, and Applications

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

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Summary

A multiple integral extends the definite integral to functions of several variables. It can measure accumulated quantities over regions in a plane, in space, or in higher dimensions.

A function of two variables integrated over a plane region gives a double integral. A function of three variables integrated over a region in space gives a triple integral. In nn dimensions, a multiple integral accumulates values of a function over an nn-dimensional domain.

All flashcards

Multiple integral
A definite integral of a function of several real variables over a multidimensional domain.
Riemann-sum term
The product f(Pk)m(Ck)f(P_k)m(C_k) estimates the function's contribution over one small cell.
Partition diameter
The largest diameter among all the cells in a partition.
Fubini's theorem
A theorem that permits computing a multiple integral by iterated integration under suitable conditions.
Mass from density
The mass of a body is the integral of its density over its volume.

Quiz questions

  1. For an absolutely integrable function on a product domain, what does Fubini's theorem guarantee?Answer: The iterated integrals agree with the multiple integral in either order.
  2. If a region is symmetric about the yy-axis and f(−x,y)=−f(x,y)f(-x,y)=-f(x,y), what is the integral of ff over that region?Answer: Zero, because contributions from reflected points cancel.
  3. For a planar region described by a≤x≤ba\leq x\leq b and α(x)≤y≤β(x)\alpha(x)\leq y\leq\beta(x), which iterated-integral setup matches vertical slices?Answer: ∫ab∫α(x)β(x)f(x,y) dy dx\int_a^b\int_{\alpha(x)}^{\beta(x)} f(x,y)\,dy\,dx
  4. What is the integral of a constant cc over a domain DD with measure m(D)m(D)?Answer: c m(D)c\,m(D)
  5. How is the mass of a body with density ρ\rho distributed through volume VV calculated?Answer: Integrate the density over the body's volume.

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

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