Question

Suppose that f ( x , y ) = x + 7 y and the region D is given by { ( x , y ) ∣ − 2 ≤ x ≤ 3 , − 2 ≤ y ≤ 3 } . D [Graphs generated by this script: initPicture(-6,6,-6,6),axes(),rect([-2,-2],[3,3]),stroke="red",text([1.5,1.5],"D") ] Then the double integral of f ( x , y ) over D is ∫ ∫ D f ( x , y ) d x d y =

Answer 1

To find the double integral of f(x, y) = x + 7y over the region D, integrate the function with respect to both x and y over the boundaries of the region.

Step-by-step explanation:

To find the double integral of f(x, y) over the region D, we need to integrate the function f(x, y) with respect to both x and y over the given region. The region D is a rectangle with boundaries -2 ≤ x ≤ 3 and -2 ≤ y ≤ 3. We can express the double integral as:

∫∫D f(x, y) dxdy

Since f(x, y) = x + 7y, the integral becomes:

∫∫D (x + 7y) dxdy

Expanding the integral and evaluating it for x and y within the given boundaries, we get:

∫∫D (x + 7y) dxdy = ∫-23 ∫-23 (x + 7y) dxdy

Integrating the function (x + 7y) with respect to x and then y, and evaluating it for the given boundaries, we can find the result of the double integral.