Question

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 integral. (a) ∬r f(x,y)da=∫ab∫cd f(x,y)dydx a= -1 b= 4 c= d= 2

Explanation:

Step1: Analyze the \(y -\)limits for a given \(x\)

For the lower \(y -\)limit (\(C\)), when \(x\) is in the range \([- 1,4]\), the lower boundary of the region \(R\) is the line passing through the points \((-1,-3)\) and \((4,0)\).
The slope of the line \(m=\frac{0 + 3}{4+1}=\frac{3}{5}\). Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(-1,-3)\), we have \(y+3=\frac{3}{5}(x + 1)\), so \(y=\frac{3}{5}x-\frac{12}{5}\).
For the upper \(y -\)limit (\(D\)), when \(x\in[-1,4]\), the upper boundary of the region \(R\) is \(y = 2\).

Answer:

\(C=\frac{3}{5}x-\frac{12}{5}\), \(D = 2\)