Calculus Volume 3 · free preview
Double Integrals over Rectangular Regions
Double integrals accumulate over plane regions as limits of double Riemann sums, equal to iterated integrals by Fubini's theorem, computing volumes under surfaces and averages over rectangles.
1. Double integrals as limits
Double integrals as limits
2. Fubini's theorem and iterated integrals
Fubini's theorem and iterated integrals
3. Volumes and averages
Volumes and averages
4. The integral's algebraic properties
The integral's algebraic properties
5. Double Riemann sum
A double Riemann sum approximating the integral over a rectangle sums which quantity?
6. Fubini content
Fubini's theorem states which equality for continuous f on a rectangle?
7. Volume interpretation
For f(x, y) at least 0 on R, the double integral of f is which geometric quantity?
8. Average value formula
The average value of f over R is which expression?
9. Order convenience
Fubini's theorem permits either integration order. When does choosing the order matter practically?
10. Iterated order count
An iterated integral performs how many successive integrations? Answer with a number.
11. Unit square area
The unit square has area 1. The average of f over it equals the double integral of f, no division needed. As a number, what is the area? Answer with a number.
12. Cross section slicing
Fubini's inner integral computes the cross-sectional area at each x. As two words, what are these slices? Answer like answer keys.
13. Non-continuous integrability
Fubini's theorem as stated needs continuous f. What can happen for discontinuous functions?
14. Constant integrand volume
The double integral of the constant 3 over a rectangle of area 5 equals what number? Answer with a number.
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