Final answer:
To write ∫ᵣ f dA as an iterated integral for the shaded region R, you need to express the differential area element in terms of Cartesian coordinates. The outer integral will have limits a to b, which correspond to the x-values where the region R starts and ends. The inner integral will have limits g(x) to h(x) for each x in the interval [a, b], which correspond to the y-values bounded by the curves g(x) and h(x).
Step-by-step explanation:
To write ∫ᵣ f dA as an iterated integral for the shaded region R, we need to express the differential area element in terms of Cartesian coordinates. Let's say the region R is bounded by the curves y = g(x) and y = h(x), where g(x) ≤ h(x) for all x in the interval [a, b]. The differential area element, dA, can be expressed as dA = dx dy or dA = dy dx, depending on the orientation.
To find the iterated integral, we need to determine the limits of integration based on the given region R. The outer integral will have limits a to b, which correspond to the x-values where the region R starts and ends. The inner integral will have limits g(x) to h(x) for each x in the interval [a, b], which correspond to the y-values bounded by the curves g(x) and h(x).
Finally, we can write the iterated integral as ∫ᵣ f dA = ∫︁︁︁︁a︁︁︁︁ b ∫︁︁︁︁g(x)︁︁︁︁︁ h(x) f(x, y) dy dx or ∫︁︁︁︁c︁︁︁︁ d ∫g(y)︁︁︁︁ h(y) f(x, y) dx dy, depending on the orientation of the differential area element.