Question

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 corners and (b) the upper left corners of the rectangles.

Answer 1

Sure, let's proceed step by step.

1. Divide the rectangle R into smaller rectangles. Since we are given that m=2 and n=3, we divide the interval [0, 4] into 2 parts and [-1, 2] into 3 parts.

2. Hence the lengths of the sides of the small rectangles are 4/2=2 and 3/1=3.

3. Now we calculate the areas of the rectangles, which are `dx*dy` = 2*3 = 6.

4. Then, calculate the sample points. For (a) the lower right corners and (b) for the upper left corners. We get these points by applying the formula:
For (a): For the interval [0,4], the sample points will be 0,2 and for [-1,2], the sample points will be -1,0,1.
For (b): For the interval [0,4], the sample points will be 2, 4 and for [-1,2], the sample points will be 0,1,2.

5. Next, apply these points to f(x, y) = 1 - x*y^2.

6. For (a), the total Riemann sum is the sum of all the values of the function at all sample points multiplied by the area of the rectangles, or `sum_a`. This equals 4.

7. For (b), the total Riemann sum is the sum of all the values of the function at all sample points multiplied by the area of the rectangles, or `sum_b`. This equals -48.

So, the estimates for the given integral, using the lower right corners and upper left corners of the rectangles as the sample points, are 4 and -48 respectively.