Let's analyze the region \( R \) step by step. ### Step 1: **Identify the region \( R \)** The shaded region is a right triangle with vertices at \((-3, 3)\), \((3, -3)\), and \((-3, -3)\). #### The boundaries: - **Left and right:** \(x\) ranges from \(-3\) to \(3\). - **Bottom:** \(y = -3\). - **Top:** The line from \((-3, 3)\) to \((3, -3)\), which is \(y = -x\). --- ## (a) \(\iint_R f(x, y) dA = \int_A^B \int_C^D f(x, y) dy\,dx\) This means **outer integral is over \(x\)**, and for each \(x\), **inner is over \(y\)**. - \(x\) ranges from \(-3\) to \(3\). - For each \(x\), \(y\) ranges from the bottom (\(y = -3\)) up to the line (\(y = -x\)). So: - \(A = -3\) - \(B = 3\) - \(C = -3\) - \(D = -x\) --- ## (b) \(\iint_R f(x, y) dA = \int_E^F \int_G^H f(x, y) dx\,dy\) This means **outer integral is over \(y\)**, and for each \(y\), **inner is over \(x\)**. - \(y\) ranges from \(-3\) to \(3\). - For each \(y\), \(x\) ranges from the left edge (\(x = -3\)) up to the line (\(x = -y\)). So: - \(E = -3\) - \(F = 3\) - \(G = -3\) - \(H = -y\) --- ## **Final Answers** **(a)** - \(A = -3\) - \(B = 3\) - \(C = -3\) - \(D = -x\) **(b)** - \(E = -3\) - \(F = 3\) - \(G = -3\) - \(H = -y\) --- ### **Summary Table** | | Lower Limit | Upper Limit | |------|-------------|-------------| | (a) | \( x: -3 \) | \( x: 3 \) | | | \( y: -3 \) | \( y: -x \) | | (b) | \( y: -3 \) | \( y: 3 \) | | | \( x: -3 \) | \( x: -y \) | Let me know if you need the integrals written out!