The contour map shows the temperature, in degrees...

The contour map shows the temperature, in degrees Fahrenheit, at 4:00 pm on February 26, 2007, in Colorado. (The state measures 388 mi west to east and 276 mi south to north.) Use the Midpoint Rule with $ m = n = 4 $ to estimate the average temperature in Colorado at that time.

Key Concepts

Average Value of a Function over a Region

The average value of a function on a region is calculated by dividing the total integral of the function over the region by the area of the region. This concept is important in applications where one needs to compute a representative value of a variable, such as temperature, pressure, or elevation, over a spatial domain.

Double Integral Approximation

A double integral represents the accumulation of a function's values over a two-dimensional region. Approximating a double integral via Riemann sums involves summing the product of function values at sample points and the area of the subregions. This approach is essential in estimating total quantities or averages over an area when an exact integral is difficult or impossible to compute analytically.

Midpoint Rule

The Midpoint Rule is a numerical integration technique that approximates the value of an integral by using the function's value at the midpoint of each subregion. In the context of double integrals, the rule involves dividing the region into a grid of subrectangles and evaluating the function at the center of each rectangle, then summing these contributions multiplied by the area of a single subrectangle.

Partitioning the Domain

Partitioning the domain involves dividing a larger area into smaller, manageable subregions (often equally sized rectangles) to simplify the process of integration or summation. The dimensions of these subregions are determined by dividing the total range by the number of subdivisions in each direction, which is fundamental in applying Riemann sum approximations like the Midpoint Rule.

Transcript

00:01 We have this contour map shown in the book, and it shows the temperature in degrees fahrenheit at some time during some day in colorado.

00:08 And the state measures 388 miles east to west and 276 miles south to north.

00:17 So what i've drawn here is that grid.

00:21 So we're going to be using the midpoint rule with n and m equal to 4 to estimate this average temperature.

00:28 Okay, so in your book you can see that we have these different contours going on.

00:34 I'm not going to draw them.

00:35 It's quite complicated, and it's going to take too much time to draw them, so i please ask you to refer to the book.

00:43 But these are the points that we are going to be interested in.

00:49 So these are all the midpoints of these squares, these rectangles.

00:58 And we want to figure out what these points are.

01:03 We want to figure out what these values are.

01:05 So we refer, again, back to the contour map to figure out these values.

01:10 So the average is going to be one over the area of the region, over the double integral, over the region of these different values, at these different points i comma j and da.

01:30 Sorry, that should be, yes, the da.

01:36 Okay, so we're going to approximate this using a remand sum.

01:40 So the remand sum, again, we still have one over the area of the region, and we've got this remand sum over i and j over these values of i, comma, j, and this delta a.

01:58 Okay, so first we'll let's figure out this delta a.

02:00 So this delta a is going to be, it's going to be, let's see, we need to figure out what the length of these size of the rectangle are.

02:16 So here we're going to have 97 and here we're going to have 69.

02:21 Okay.

02:22 So our delta area, if we multiply these two guys together, we're going to get 6 ,6, 9.

02:29 Okay? so this is going to be one over the area of the region...

Please give Ace some feedback