Question

If R=[-1,3] ×[4,6], use a Riemann sum with m=4, n=2 to estimate the value of the following.

∬ᵣ(y²-2 x²) d A
Take the sample points to be the upper left corners of the squares. The value of integral is__

Answer 1

To estimate the integral using a Riemann sum with m=4 and n=2, divide the region R=[-1,3]x[4,6] into rectangles, evaluate the function value at the upper left corners of each subrectangle, and sum the products of these values with the respective rectangle areas.

Step-by-step explanation:

To estimate the value of the integral ∫∫_R(y² - 2x²) dA using a Riemann sum with m=4 and n=2, we first define our region R=[-1,3] × [4,6]. Since m=4, this means we divide the x-interval [-1,3] into 4 equal parts, each with a width of Δx=(3-(-1))/4=1. Similarly, with n=2, we divide the y-interval [4,6] into 2 parts with a height of Δy=(6-4)/2=1. The corners of the subintervals will be our sample points.

For each subrectangle with corners at (x_i,y_j), the Riemann sum will be the sum of the areas multiplied by the function value at the upper left corner of each subrectangle. Thus, we calculate the Riemann sum as:

Σ_{i=1}^{4}Σ_{j=1}^{2} (f(x_i^*, y_j^*) Δx Δy) where (x_i^*, y_j^*) are the upper left corners. Plug the coordinates of these points into the function and multiply by the area of each subrectangle (Δx*Δy).

Once the sum is calculated, it provides an estimate of the double integral over the region R.