Question

express the double integral d f (x, y) da as an iterated integral for the given function f and region d.

Answer 1

To convert a double integral into an iterated integral, determine the integration limits for the variables x and y within the region D, choose an order of integration, and then carry out the integration process.

Step-by-step explanation:

To express the double integral d f (x, y) da as an iterated integral for a given function f and region D, one must first define the limits of integration for the variables x and y within the region D. These limits will be dependent on the shape of the region D. Once the limits are identified, the double integral can be written as an iterated integral, integrating first with respect to y, and then with respect to x, or vice versa. The integration order is chosen based on the simplicity of the functions involved and the geometry of the region D.

For example, if the region D is a circle of constant radius r, one might choose polar coordinates where dA = r dθ dr, and thus integrate first over the angle θ, and then over the radius r. Alternatively, if D is rectangular, the limits can be constant and the integral simplified. The key is setting up the iterated integral correctly and then performing the actual integration.

Answer 2

To express the double integral d f (x, y) da as an iterated integral, determine the limits of integration for both the x and y variables and define the region D. Then, write the double integral as an iterated integral with the inner integral representing the integrated variable and the outer integral representing the integrating variable.

Step-by-step explanation:

To express the double integral ∫∫ f(x, y) da as an iterated integral, we need to determine the limits of integration for both the x and y variables. The region D can be defined by specifying the range of values for x and y that make up the region. Once we have the limits of integration, we can write the double integral as an iterated integral with the inner integral representing the integrated variable and the outer integral representing the integrating variable.

For example, if the region D is defined by x2 + y2 ≤ 1, we can express the double integral as:

∫∫ f(x, y) da = ∫[lower limit y to upper limit y] ∫[lower limit x to upper limit x] f(x, y) dx dy