Question

Calc III Tutorial: Iterated Integrals

Answer 1

\({\bf{Double~Integrals:}}\)
consider a rectangle on the xy plane, defined by

a <= x <= b
c <= y <= d

that is sub-divided into smaller rectangles by some number of horizontal and vertical lines. the area of the region is the sum of these rectangles.

let the width of each smaller rectangle be Δx and the height be Δy. the area of each partition is naturally ΔA = ΔxΔy. we can pick some arbitrary point xk yk and evaluate the function at that point. then, we can multiply that value by ΔA or ΔxΔy to get the volume contained between the xy plane and the function.

summing all such volumes gets an approximation of the total volume.

\[S_{n}=\sum_{k=1}^{n}f(x_{k}, y_{k})ΔxΔy\]

let the norm be ||P||, the maximum height and width of all the rectangles. as the norm approaches 0, the area of each rectangle approaches 0. if we take the sum of the rectangles as norm approaches 0, we get the volume of the region.



this situation is analogous to single variable Riemann sums, where you divided a function into a series of rectangles and took the integral as the sum of such rectangle areas as their widths approached 0.

Answer 2

when the limit Sn exists such that the limit is the same regardless of which xk,yk is chosen, then the function is integrable and the limit can be written as a double integral of f over R

\[\int\limits_{}^{} \int\limits_{R}^{}f(x,y)dA\]
where one of the integrals corresponds to the x-boundaries and the other integral corresponds to the y-boundary. to evaluate a double integral, start with the first variable listed in the differentiation part. for example, if the integral is

\[\int\limits_{x1}^{x2} \int\limits_{y1}^{y2}f(x,y)dydx\]
you would start by integrating wrt y first , keeping x constant. then, after the first integration step is complete, integrate wrt to x.
by convention, the boundaries of the integral are written in the opposite order in which you are supposed to differentiate them.

Answer 3

\({\bf{Fubini's~Theorem}}\): states that you can switch the order of integration and obtain the same result. that is,

\[\int\limits_{y1}^{y2}\int\limits_{x1}^{x2}f(x,y)dxdy = \int\limits_{x1}^{x2}\int\limits_{y1}^{y2}f(x,y)dydy \]

so if you find that one particular double integral ends up being unsolvable/difficult to solve by hand, try switching the order of integration.

Answer 4

Source material is section 15.1 of Thomas' Calculus, Early Transcendentals, 14th edition by Hass, Heil, Weir, et. al.