Double integrals beyond volume

Kontekst

• Double integrals over non-rectangular regions

Do czego zmierzamy

• Double integrals are used anytime you get that feeling where you want to chop up a two-dimensional region into infinitely many infinitely small areas, multiply each one by some value, then add them up.
• The more general notation for a double integral is
  $\begin{aligned} \iint_\blueE{R} f\,\redE{dA} \end{aligned}$
  where
• $\blueE{R}$start color #0c7f99, R, end color #0c7f99 is the region that you are integrating over.
• $\redE{dA}$start color #bc2612, d, A, end color #bc2612 signifies a "tiny chunk of area", which typically means $dx\,dy$d, x, d, y or $dy\,dx$d, y, d, x, unless another coordinate system is being used.
• $f(x, y)$f, left parenthesis, x, comma, y, right parenthesis is a two-variable function.

Example 1: Mass of a plate

Wybierz 1 odpowiedź:

Wybierz 1 odpowiedź:

• (Choice A)  

  $\begin{aligned} \int_0^3 \int_0^2 \sigma(x, y)\,dx\,dy \end{aligned}$

  A

  $\begin{aligned} \int_0^3 \int_0^2 \sigma(x, y)\,dx\,dy \end{aligned}$
• (Choice B)  

  $\begin{aligned} \int_0^2 \int_0^3 \sigma(x, y)\,dx\,dy \end{aligned}$

  B

  $\begin{aligned} \int_0^2 \int_0^3 \sigma(x, y)\,dx\,dy \end{aligned}$

Sprawdź

[Odpowiedź]

$\begin{aligned} \int_0^2 \int_0^3 (\sin(\pi x) + 1)y \,dx\,dy = \end{aligned}$ =

Sprawdź

[Odpowiedź]

Thinking about tiny areas

• First cut the volume into infinitely many slices. Each slice represents a constant value for one of the variables, for example $x = 0.78$x, equals, 0, point, 78.
• Find the area of each of those slices. (This is what the inner integral does).
• Make each slice an infinitesimal volume by giving it a little depth. Mathematically, this means multiplying the area of each slice by either $dx$d, x or $dy$d, y, whichever one represents a tiny step perpendicular to the slice.
• Integrate those infinitesimal volumes together to get the volume of the solid as a whole. (This is what the outer integral does).

General notation for double integrals

• $\blueE{R}$start color #0c7f99, R, end color #0c7f99 represents the region that we are integrating over. The reason for writing it like this is that while you are setting things up, or reasoning about double integrals in general, you typically don't want to get bogged down with the specific (and potentially complicated) definition of your region while you scribble things down.
  When it comes time to compute the integral, we replace this $\displaystyle \iint_\blueE{R}$\iint, start subscript, start color #0c7f99, R, end color #0c7f99, end subscript with an actual pair of single-integrals with bounds that can be computed. When $\blueE{R}$start color #0c7f99, R, end color #0c7f99 is a rectangle, those bounds will be constants:

  $\displaystyle \int_{y_1}^{y_2} \int_{x_1}^{x_2} \dots$integral, start subscript, y, start subscript, 1, end subscript, end subscript, start superscript, y, start subscript, 2, end subscript, end superscript, integral, start subscript, x, start subscript, 1, end subscript, end subscript, start superscript, x, start subscript, 2, end subscript, end superscript, dots

  More generally, when $\blueE{R}$start color #0c7f99, R, end color #0c7f99 is defined in terms of some curves in the $xy$x, y-plane, the bounds of the inner integral are expressed as functions of the outer variable:

  $\displaystyle \int_{y_1}^{y_2} \int_{x_1(y)}^{x_2(y)} \dots$integral, start subscript, y, start subscript, 1, end subscript, end subscript, start superscript, y, start subscript, 2, end subscript, end superscript, integral, start subscript, x, start subscript, 1, end subscript, left parenthesis, y, right parenthesis, end subscript, start superscript, x, start subscript, 2, end subscript, left parenthesis, y, right parenthesis, end superscript, dots

  (See the last article for practice with this idea.)
• $\redE{dA}$start color #bc2612, d, A, end color #bc2612 represents a tiny area, in the same way that $dx$d, x represents a tiny length in an ordinary integral.
  You will typically imagine chopping up the region $\blueE{R}$start color #0c7f99, R, end color #0c7f99 into many tiny pieces, and this term represents the area of one of those pieces. Once you get down to computing the double integral, you will replace this with $dx\,dy$d, x, d, y, or $dy\,dx$d, y, d, x. In other coordinate systems, there are different ways to break down $\redE{dA}$start color #bc2612, d, A, end color #bc2612, but I'll leave that for the next article.
• $f(x, y)$f, left parenthesis, x, comma, y, right parenthesis is some two-variable function. When you chop up your region into many tiny pieces, each piece typically represents some value that you are hoping to add up. Perhaps this value is a tiny bit of mass, or the tiny volume of a slim column under a graph. Hopefully, you are able to express this tiny amount as something times the area of your tiny piece. For example, the mass of a piece is its density times its area; and the volume of a column above a piece is the height of the column times the area.
  In these examples, $f(x, y)$f, left parenthesis, x, comma, y, right parenthesis represents density, or height. In general, it is the thing that needs to be multiplied by the area $\redE{dA}$start color #bc2612, d, A, end color #bc2612 of a tiny piece, and it generally depends on the position of that tiny piece, expressed with $(x, y)$left parenthesis, x, comma, y, right parenthesis-coordinates.

  "What if I cannot express the tiny value that I want to add up as something times $dA$d, A?"

  Well, in that case my friend, double integrals are not the tool for you. Although I cannot think of any examples where that comes up...

• Simplicity: When you're starting to set something up, or if you want to quickly reference the idea of a certain double integral without getting into the implementation details, it's nice to be able to write something quickly. Also, many of the theorems and tools coming up in multivariable calculus are expressed abstractly in this notation.
• Generality: Writing your integral as $\displaystyle \iint_\blueE{R} f\,\redE{dA}$\iint, start subscript, start color #0c7f99, R, end color #0c7f99, end subscript, f, start color #bc2612, d, A, end color #bc2612 gives you options as you sit down to compute it. For example, in the next article, we will cover double integrals in polar coordinates, in which case the way you expand $\redE{dA}$start color #bc2612, d, A, end color #bc2612 and the way you put bounds on the two integrals are different than they are for cartesian coordinates.

Example 2: Center of mass

Podsumowanie

• Double integrals are used anytime you get that feeling where you want to chop up a two-dimensional region into infinitely many infinitely small areas, multiply each one by some value, then add them up.
• The more general notation for a double integral is
  $\begin{aligned} \iint_\blueE{R} f\,\redE{dA} \end{aligned}$
  where

  • $\blueE{R}$start color #0c7f99, R, end color #0c7f99 is the region that you are integrating over.
  • $\redE{dA}$start color #bc2612, d, A, end color #bc2612 signifies a "tiny chunk of area", which typically means $dx\,dy$d, x, d, y or $dy\,dx$d, y, d, x, unless another coordinate system is being used.
  • $f(x, y)$f, left parenthesis, x, comma, y, right parenthesis is a two-variable function.
• From this point forward, double integrals will be inextricably tied to most of the new topics in multivariable calculus. And in almost all cases, it helps to think about what's happening inside each "tiny area" of a given region, rather than thinking about integrating something along a line then integrating again in the perpendicular direction.