Let's break down the **iterated integral** and describe the region and its sketch step by step. ### 1. **Understanding the Integral** \[ \int_0^1 \int_0^{1-x} \int_0^{8-8z} dy\ dz\ dx \] #### **Order of Integration:** - **Inner:** \( y \) from \( 0 \) to \( 8 - 8z \) - **Middle:** \( z \) from \( 0 \) to \( 1-x \) - **Outer:** \( x \) from \( 0 \) to \( 1 \) --- ### 2. **Describing the Region** #### **Step 1: Outer Integral (\(x\))** - \(0 \leq x \leq 1\) #### **Step 2: Middle Integral (\(z\))** - \(0 \leq z \leq 1 - x\) #### **Step 3: Inner Integral (\(y\))** - \(0 \leq y \leq 8 - 8z\) --- ### 3. **Visualizing the Bounds** #### **Base in the \(xz\)-plane:** - For the base, set \(y = 0\): - \(0 \leq x \leq 1\) - \(0 \leq z \leq 1-x\) - This is a **triangle** in the \(xz\)-plane with vertices at \((0,0), (1,0), (0,1)\). #### **Top Surface:** - The upper bound for \(z\) is \(1-x\), which is a plane sloping down as \(x\) increases. - The upper bound for \(y\) is \(8-8z\), which is a plane sloping down as \(z\) increases. #### **Faces:** - **In the \(xz\)-plane:** Triangular face (since bounds are linear in both \(x\) and \(z\)) - **In the \(yz\)-plane:** Triangular face (since \(y\) goes from \(0\) to \(8-8z\), which is linear in \(z\)) - **Plane \(z = 1-x\):** Triangular face (as \(z\) decreases with increasing \(x\)) - **Plane \(y = 8-8z\):** Triangular face (as \(y\) decreases with increasing \(z\)) --- ### 4. **Region Description** - The **solid has a triangular base** in the \(xz\)-plane. - The **solid has triangular faces** in the \(xz\)-, \(yz\)-planes, in the plane \(z=1-x\), and in the plane \(y=8-8z\). - As **\(x\) increases, the top of the region decreases** (since \(z\) max decreases). - As **\(y\) increases, the top of the region decreases** (since \(y\) max decreases as \(z\) increases). --- ### 5. **Final Answers (Based on Choices in the Image)** - **The solid has a triangular face in the xz-plane.** ✅ - **The solid has a triangular face in the yz-plane.** ✅ - **The solid has a triangular face in the plane \(z = 1-x\).** ✅ - **The solid has a triangular face in the plane \(y = 8-8z\).** ✅ - **As \(x\) increases, the top of the region decreases.** ✅ - **As \(y\) increases, the top of the region decreases.** ✅ The only incorrect option is: - **The solid has a trapezoidal base in the xy-plane.** ❌ (It should be a triangular base in the xz-plane.) --- ## **Summary** - The base is a triangle in the \(xz\)-plane. - All faces described are triangular. - As either \(x\) or \(y\) increases, the "top" of the region decreases. - The solid is a wedge-like shape bounded by planes. **Let me know if you want a sketch or further clarification!**