Suppose R is the shaded region in the figure, and f(x, y) is a continuous function on R. Find the limits of integration for the following iterated integrals. (a) ff f(x, y) dA = f(x, y) dy dz R

Let f(x, y) = x²e² and let R be the triangle bounded by the lines x = 4, x = y/2, and y = x in the xy-plane. (a) Express SR f dA as a double integral in two different ways by filling in the values for the integrals below. (For one of these it will be necessary to write the double integral as a sum of two integrals, as indicated; for the other, it can be written as a single integral.) JRf dA= f f f(x,y) d where a = , C = ,C= And fRf dA= f f f(x, y) d + Sm Sp f(x,y) d d where a = 1 m = ,b= ,p= = ,b= d d and d = ,d= n = , and q = (b) Evaluate one of your integrals to find the value of SR f dA. SR f dA=

Evaluate the integral of the function f(x, y) over the region R that is bounded by the x-axis, the line x = 4, and the curve y = sqrt(x). Set up integrals with both dA = dx dy and dA = dy dx.

x F. dr, where F(x, y) = ( Y y+3 x 4, 4).0 C is the path along (x − 5)² + (y + 3)² = 4, traversed counter-clockwise exactly once, starting at (7, -3) and ending at (7,-3). Determine whether the circulation form of Green's Theorem can be directly applied to evaluate this integral. " Consider the line integral F Select the correct answer below: Green's Theorem does not apply because the path C' is not closed. Green's Theorem does not apply because F(x, y) does not have continuous partials in the interior of C. O Green's Theorem can be directly applied to evaluate this integral.