If R=[0,4] ×[-1,2], use a Riemann sum with m=2 n=3 to...

If $R=[0,4] \times[-1,2],$ use a Riemann sum with $m=2$ $n=3$ to estimate the value of $\iint_{R}\left(1-x y^{2}\right) d A$ . Take the sample points to be (a) the lower right comers and (b) the upper left corners of the rectangles.

Key Concepts

Double Integration

Double integration extends the concept of integration to functions of two variables over a two-dimensional region. It is used to compute quantities such as area, volume under a surface, and other aggregated properties across a region, and it involves iterated integration over two variables.

Riemann Sum Approximation

A Riemann sum is a method for approximating the value of a double integral by summing over a finite number of subregions. In each subregion, the function is evaluated at a selected point and multiplied by the area of the subregion. As the number of subdivisions increases, the Riemann sum converges to the exact value of the integral.

Partitioning of the Integration Region

The partitioning of a region involves dividing the overall integration region into smaller subrectangles (or subregions). For rectangular regions, this is done by uniformly subdividing the ranges of each variable, allowing for a structured approach to approximating an integral over the domain.

Sample Point Selection

The choice of sample points in each subrectangle, such as selecting the lower right or upper left corners, determines where the function is evaluated. This selection can affect the estimate obtained by the Riemann sum, as different points may yield higher or lower contributions depending on the function's behavior over the subrectangle.

Subrectangle Area Calculation

Each subrectangle in the partition has an area calculated as the product of its side lengths. This area is used as a weight in the Riemann sum, multiplying the function’s value at the chosen sample point to approximate the contribution of that subregion to the overall integral.

Transcript

00:01 In this problem, we're going to estimate the volume under this surface, 1 minus x, y squared, over this rectangular region, x from 0 to 4, and y from negative 1 to 2.

00:12 And we're going to make 6 rectangles.

00:15 You can see them here below.

00:17 The first time through, we're going to use the lower right -hand corner to find the height of the rectangular region underneath there.

00:28 Okay, lower right for the first corner is right here, which would be negative 1, 2.

00:42 Oh, that's just backwards.

00:46 2 negative 1.

00:52 Then the next one would be 4 negative 1.

01:00 Then the next one would be 2 .0 and 4 .0.

01:04 And then the next one would be 2 -1 and 4 -1.

01:24 Then to find the height, the z, we just plug into this function right here.

01:28 So square the y, multiply by x, and then do 1 minus that.

01:33 I get negative 1 there.

01:37 It's 4, negative 3, 1, 1, 1, 1 minus 2, 1, 1 minus 4, 4, 4, dx, well the width of the x, the x width of the rectangles here is two for all of them.

02:05 D .y, that's from here to here, and that's one.

02:13 And volume is height times width, times length.

02:18 So negative 2, negative 6, 2, negative 2, negative 6...

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