Integration

startsection section10mm-.5 Discussion 28 Total Accumulation

 

In this discussion we want to develop a method for approximating the total accumulation of a function of two variables. The particular function that we will be using is ``daily snowfall'' and the total accumulation will be the total volume of snowfall.

On March 12-15, 1993, the so-called ``Storm of the Century'' hit the east coast of the United States. Snowfall totals over 50 inches were recorded in some areas of Tennessee and North Carolina with drifts of over 14 feet. The National Weather Service's Office of Hydrology estimated the volume of water that fell as snow as 44 million acre-feet, which is comparable to 40 days' flow on the Mississippi River at New Orleans. (An acre-foot is the amount of water it would take to cover one acre to a depth of one foot.) In this discussion, using data and a plot from the National Climatic Data Center, we would like to determine the volume of water in the form of snow that fell on one region--New Jersey, New York (excluding Long Island), and Pennsylvania--on one day of this storm. The ratio of snow depth to water depth was approximately 5:1, so that 5 inches of snow is equivalent to 1 inch of rain. Thus the total volume of water that fell will be 1/5 the total volume of snow that fell.

startsection subsection10mm.5 Exercises Figure contains the snowfall data for March 13, 1993 for New Jersey, New York, and Pennsylvania. The locations of the data collection sites are given in latitude and longitude and the snowfall measurements are given in tenths of inches. The locations are marked on the density plot in Figure . The plot is shaded according to the accompanying key to indicate snowfall amounts for March 13, 1993, for the eastern United States. Note: There are 640 acres in a square mile, one degree of latitude is approximately 69 miles, and one degree of longitude (in this region) is approximately 51 miles. Your numerical answers should be given in square mile-feet of snow.

1.

(a)

Given a region of the plot in Figure what method would you use to approximate the total volume of snow that fell on the region?

(b)

Use your method to produce a value that approximates the total volume of snow that fell on New Jersey, New York (excluding Long Island), and Pennsylvania.

2.

(a)

How would you improve upon your method to produce a better approximation to the total volume of snow? Explain why this would produce a better approximation to the total volume of snow.

(b)

Based upon your answer to Exercise 2(a), improve upon the approximation that you produced in Exercise 1(b).

3.

Can your method be used to produce an exact value for the total volume of snow represented by the plot? If so, explain how this might be done. In particular, what, if any, additional information about the snowfall function would you need to know in order to produce an exact value by this method.

  

Figure: Data for snowfall in the New Jersey, New York, and Pennsylvania on March 13, 1993, from the ``Storm of the Century''. The amount of snow is given as a function of latitude and longitude and is given in tenths of inches. Note that west longitude is written as a negative number. (Data is courtesy of the National Climatic Data Center.)

  

Figure:

startsection section10mm-.5 Discussion 29 Iterated Integrals in the Plane

 

In this discussion we would like to focus on the connection between partitions of a region ${\cal R}\subset {\bf R}^2$ and the iterated integral representation of a double integral over ${\cal R}$, which is given by Fubini's theorem (see Section 5.2 of the text). Let us first review this connection.

Suppose that ${\cal R}$ is defined by the inequalities $a \leq x \leq b$ and $g_{1}(x) \leq y \leq g_{2}(x)$, where g₁ and g₂ are continuous functions of one variable. In order to partition ${\cal R}$ into $M\times N$ rectangular cells, we first partitioned ${\cal R}$ into M columns of width $\Delta x_i$, $i =1,\ldots, M$, and then partitioned the i^th column into N cells of height $\Delta y_{ij}$. (See Figure .) This led to a Riemann sum for the total accumulation of f which could be written as an iterated sum,

$$R(f,P) = \sum_{i=1}^{M} \left(\sum_{j=1}^{M} f(x_{i}^{*},y_{ij}^{*}) \Delta y_{ij}\right) \Delta x_i,$$

where we sampled f in the ij^th cell at (x_i^*, y_ij^*). The limit of Riemann sums of this form is the iterated integral

$$\int_{a}^{b}\int_{g_{1}(x)}^{g_{2}(x)} f(x,y)\,dydx.$$

The order of evaluation of the iterated integral, first y and then x, is determined by the form of the Riemann sum, which is in turn determined by the geometry of ${\cal R}$.

    

Figure: (a) A region defined by $a \leq x \leq b$ and $g_{1}(x) \leq y \leq g_{2}(x)$ partitioned into M=6 columns with N=5 cells in each column. The second cell in the third column, which has dimensions $\Delta x_{3}\times \Delta y_{32}$, is shaded. (b) A region defined by $c \leq y \leq d$ and $g_1(y) \leq x \leq g_{2}(y)$partitioned into N=5 rows with M=3 cells in each row. The second cell in the second row, which has dimensions $\Delta x_{22}\times \Delta y_{2}$, is shaded.

If the roles of x and y are reversed, and ${\cal R}$ is defined by inequalities $c \leq y \leq d$ and $g_1(y) \leq x \leq g_{2}(y)$, then we would first partition ${\cal R}$ into N rows, and then partition each row into M cells. (See Figure .) After writing out the corresponding iterated sum form for the Riemann sum and evaluating the limit of Riemann sums, we arrived at the following iterated integral,

$$\int_{c}^{d} \int_a^b f(x,y)\, dxdy.$$

Here we integrate first with respect to x and then with respect to y.

To summarize: when ${\cal R}$ lies between vertical lines x=a and x=b, we first partitioned ${\cal R}$ into columns and then partitioned the columns, which ultimately gives an iterated integral first with respect to y and then with respect to x; and when ${\cal R}$ lies between horizontal lines y = c and y = d, we first partitioned ${\cal R}$ into rows and then partitioned the rows, which gives an iterated integral first with respect to x and then with respect to y.

startsection subsection10mm.5 Exercises

1.

(a)

For each of the regions in Figure , sketch a partition of the region consisting of $4 \times 5$ rectangular cells organized either into columns or into rows. Explain your choice.

(b)

Set up the iterated integral of a function f=f(x,y) over the regions of Figure corresponding to your partitions in part (a).

    

Figure: (a) ${{\cal R}} = \{(x,y):\,-1 \leq x \leq 1\mbox{ and } 0 \leq y \leq 2 - x^2\}$. (b) ${{\cal R}} = \{(x,y):\, 1 \leq y \leq 2, \, 1\leq x \leq y^2 \}$.

2.

For each of the following regions ${\cal R}$, (i) Sketch ${\cal R}$ and a partition of ${\cal R}$ into rectangular cells organized either into columns or into rows. Be sure to label the boundary curves of ${\cal R}$. (ii) Set up the iterated integral of a function f over ${\cal R}$ corresponding to your partition.

(a)

${\cal R}= \{(x,y): \,x^2 + y^2/4 \leq 1\}$.

(b)

${\cal R}= \{(x,y): \, x \leq 1 - y^2 \mbox{ and } x \geq y^2 - 1\}.$

(c)

${\cal R}=\{(x,y): \, y^2 \leq x^2 + 1 \mbox{ and } \vert x\vert \leq 2\}.$

3.

For each of the regions in Exercises 1 and 2, if possible, describe the region so that the roles of x and y are reversed from the original description, sketch a partition of the region into columns of equal width or rows of equal height corresponding to the new description, and give the corresponding iterated integral. If this is not possible, explain why not.

startsection section10mm-.5 Discussion 30 Integration in Polar Coordinates

 This discussion provides an opportunity to work through several examples of integration in the plane using polar coordinates and to think about the examples where it is necessary to choose an appropriate coordinate system before carrying out an integration. coordinates.

startsection subsection10mm.5 Exercises

1.

Let ${\cal R}$ be the region in the first quadrant of the xy-plane that lies between the circle of radius 1 centered at the origin and the circle of radius 2 centered at the origin. Let ${\cal D}$ be the region in the first octant of space lying over ${\cal R}$and under the graph of the function f(x,y) = 8 - x² -y².

(a)

Sketch the region ${\cal R}$ and a partition of ${\cal R}$ into polar rectangles.

(b)

Set up and evaluate a polar integral for the volume of the region ${\cal D}$.

2.

A plane carrying a load of 500 killer bees has conveniently crashed at the origin in the xy-plane. Over time, the bees spread out from the origin in a random manner so that the concentration of bees per square mile can be modelled by a function c of x, y, and t,

$$c(x,y,t) = \frac{500}{4\pi t}e^{\frac{-(x^2+y^2)}{4t}}.$$

Here t=0 corresponds to the time of the crash. Fixing t, c(x,y,t) represents the concentration of bees per square mile at time t at the point (x,y). We are interested in finding the number of bees that remain within one mile from the origin after 1/2 hour.

(a)

Sketch a partition of the region ${\cal R}$ consisting of points of distance less than or equal to 1 from the origin. Use $6 \times 5$ polar rectangles.

(b)

Set up and evaluate a polar integral for the total number of bees that remain inside the area consisting of points of distance less than or equal to 1 mile from the origin after 1/2 hour.

3.

For each of the regions ${\cal R}$ and functions f, (i) sketch the region ${\cal R}$, (ii) partition ${\cal R}$ using either rectangular or polar coordinates, (iii) set up and evaluate an integral for the total accumulation of f on ${\cal R}$ corresponding to your choice of partition of ${\cal R}$ .

(a)

${\cal R}$ is the set of points (x,y) in the first quadrant of the xy-plane satisfying $y \geq x$ and $x^2 +y^2 \leq 1$. f(x,y) = y.

(b)

${\cal R} = \{(x,y) : \ x^2 +y^2 \leq 4 \mbox{ and } y \geq 1\}$. f(x,y) = x² y.

(c)

${\cal R} = \{(x,y) : \ y \geq \vert x \vert \mbox{ and } y \leq \sqrt{4-x^2}\}.$ f(x,y) = x.

startsection section10mm-.5 Discussion 31 Iterated Integrals in Space

 

In this discussion we would like to focus on the connection between partitions of a region ${\cal R}\subset {\bf R}^3$ and the iterated integral representation of the integral of a function over ${\cal R}$. As was the case in the plane, this is given by Fubini's theorem. In particular, we would like to introduce an intuitive way to think about this connection in terms of sampling schemes. Let us work with a region ${\cal R}$ defined by $a \leq x \leq b$, $g_{1}(x) \leq y \leq g_{2}(x)$, and $h_{1}(x,y) \leq z \leq h_{2}(x,y)$, where g₁, g₂, h₁ and h₂ are continuous functions.

In order to construct a Riemann sum for the total accumulation of fon ${\cal R}$, we will sample the value of f by inserting ``sensors'' on ``wires'' into ${\cal R}$ to record the values of f. For example, if ${\cal R}$ represents a container of liquid and we want to find the total heat of the liquid, we would insert $L \times M$ wires into the fluid with N small temperature sensors on each wire to measure the temperature in the container. Intuitively, we would insert the wires parallel to the z-axis through the top of the container in a rectangular pattern and the space the N sensors along the wires. The pattern of the wires depends on the geometry of the projection $\tilde{{\cal R}}$ of the region in the xy-plane. We would partition $\tilde{{\cal R}}$ into $L \times M$ cells as we did in Discussion 25, and then insert one wire into each cell. In this case, $\tilde{{\cal R}} = \{(x,y):\,a \leq x \leq b \mbox{ and }g_{1}(x) \leq y \leq g_{2}(x)\}$. (See Figure .) Since $\tilde{{\cal R}}$ lies between the graphs of functions of x, over an interval $a \leq x \leq b$, we would partition $\tilde{{\cal R}}$ into L columns with Mcells in each column and then insert the $L \times M$ wires into the cells of the partition of $\tilde{{\cal R}}$. The sensors should be placed on the wires so that they are located above the graph of h₁ and below the graph of h₂. This sampling scheme would correspond to an iterated Riemann sum of the form

$$\sum_{i=1}^{L}\left( \sum_{j=1}^{M}\left( \sum_{k=1}^N f(x_{... ...k}^*) \Delta z_{ijk}\right) \Delta y_{ij}\right) \Delta x_i,$$

where (x_i^*,y_ij^*,z_ijk^*) is the location of the ijk^th sensor and $\Delta z_{ijk}\Delta y_{ij} \Delta x_i$ is the volume of the three-dimensional cell containing this sensor. This Riemann sum corresponds to the triple iterated integral

$$\int_{a}^{b} \int_{g_1(x,y)}^{g_2(x,y)} \int_{h_1(x,y)}^{h_{2}(x,y)} f(x,y,z) \,dz dy dx.$$

Notice that the direction of the wires, parallel to the z-axis, corresponds to the inner or first variable of integration. This was determined by our description of ${\cal R}$ as lying between the graphs of two functions of x and y. The second and third variable of integration correspond to the partition of $\tilde{{\cal R}}$ and are determined by our description of $\tilde{{\cal R}}$ as lying between the graphs of two functions of x, for $a \leq x \leq b$. .

    

Figure: (a) A region ${\cal R}$ defined by $a \leq x \leq b$, $g_{1}(x) \leq y \leq g_{2}(x)$, and $h_{1}(x,y) \leq z \leq h_{2}(x,y)$. Here, $3\times 3 \times 4$sensors are placed on $3 \times 3$ wires inserted into ${\cal R}$ parallel to the z-axis. Note the partition of ${\cal R}$ is not shown. (b) The projection of ${\cal R}$ in the xy-plane is defined by $a \leq x \leq b$and $g_{1}(x) \leq y \leq g_{2}(x)$. The wires in (a) are located at the points (x_i,y_ij) in (b), which lie in the projections to the xy-plane of the cells of the partition of ${\cal R}$.

If ${\cal R}$ had been described instead as lying between the graphs of functions of y and z, we would have inserted the wires parallel to the x-axis. The location of the wires would then be determined by the description of the projection of ${\cal R}$ in the yz-plane. Similarly, if ${\cal R}$ had been described as lying between the graphs of functions of y and z, we would have inserted the wires parallel to the y-axis. The location of the wires would then be determined by the description of the projection of ${\cal R}$ in the xz-plane.

    

Figure: (a) The region ${\cal R}= \{(x,y,z):\ x^2 + y^2/4 \leq z \leq 4\, \}$ of Exercise 1. (b) The region ${\cal R}= \{(x,y,z):\ -1 \leq z \leq 1,\ z^2 -1 \leq y \leq 1 - z^2, \mbox{ and }0 \leq x \leq z +1\ \}.$ of Exercise 2.

startsection subsection10mm.5 Exercises Each of these exercises refers to the intuitive process described above for sampling a function f on a region ${\cal R}$ and writing out a corresponding iterated integral.

1.

For the region ${\cal R}$ in space described by $x^2 + y^2/4 \leq z \leq 4$ (see Figure ), describe the projection of ${\cal R}$ in the xy-plane, describe how to insert the wires into ${\cal R}$, describe the placement of the sensors on the wires, and give the corresponding iterated integral.

2.

For the region ${\cal R}$ in space described by $-1 \leq z \leq 1$, $z^2 -1 \leq y \leq 1 - z^2$, and $0 \leq x \leq z +1$ (see Figure ), choose an axis to use for the sampling process, describe the projection of ${\cal R}$ in the corresponding coordinate plane, describe how to insert the wires into ${\cal R}$, describe the placement of the sensors on the wires, and give the corresponding iterated integral.

3.

For the region ${\cal R}$ in space described by $y^2 + z^2 \leq 4$ and $y-4 \leq x \leq y+4$ , choose an axis to use for the sampling process, describe the projection of ${\cal R}$ in the corresponding coordinate plane, describe how to insert the wires into ${\cal R}$, describe the placement of the sensors on the wires, and give the corresponding iterated integral.

4.

According to our intuitive sampling scheme, we inserted wires in ${\cal R}$ parallel to a coordinate axis in an arrangement of rows or an arrangement of columns, depending on the description of the projection of ${\cal R}$ in the coordinate plane perpendicular to this axis. Let us suppose that ${\cal R}$ is the region lying between two functions of xand y and that the projections $\tilde{{\cal R}}$ of ${\cal R}$ in the xy-plane is a disk of radius r₀ centered at the origin. In this case, ignoring the idea of columns or rows, how might you insert $L \times M$ wires into ${\cal R}$ to take advantage of the geometry of $\tilde{{\cal R}}$? Justify this arrangement and describe the resulting arrangement of sensors.