Let's analyze the given integral and its region: ### Original Integral: \[ \int_0^6 \int_0^{\sqrt{y}} \int_0^{6-y} f(x, y, z) \, dz \, dx \, dy \] #### **Region Description:** - \(0 \leq y \leq 6\) - \(0 \leq x \leq y^2\) - \(0 \leq z \leq 6-y\) ### **Given Relations:** - \(y(x) = \sqrt{x}\) - \(z(y) = 6-y\) --- ## **Step 1: Find the bounds for each order** ### **Order 1: \( dz \, dx \, dy \)** Original order is \( dz \, dx \, dy \), so let's write the bounds directly: \[ \int_{y=0}^6 \int_{x=0}^{y^2} \int_{z=0}^{6-y} f(x, y, z) \, dz \, dx \, dy \] ### **Order 2: \( dz \, dy \, dx \)** - \(x: 0 \to 6\) - For a fixed \(x\), \(y: \sqrt{x} \to 6\) (since \(x \leq y^2 \implies y \geq \sqrt{x}\)) - For a fixed \(y\), \(z: 0 \to 6-y\) So, \[ \int_{x=0}^6 \int_{y=\sqrt{x}}^6 \int_{z=0}^{6-y} f(x, y, z) \, dz \, dy \, dx \] ### **Order 3: \( dx \, dz \, dy \)** - \(y: 0 \to 6\) - For a fixed \(y\), \(z: 0 \to 6-y\) - For a fixed \(y\), \(x: 0 \to y^2\) So, \[ \int_{y=0}^6 \int_{z=0}^{6-y} \int_{x=0}^{y^2} f(x, y, z) \, dx \, dz \, dy \] ### **Order 4: \( dx \, dy \, dz \)** - \(z: 0 \to 6\) - For a fixed \(z\), \(y: 0 \to 6-z\) (since \(z = 6-y \implies y = 6-z\)) - For a fixed \(y\), \(x: 0 \to y^2\) So, \[ \int_{z=0}^6 \int_{y=0}^{6-z} \int_{x=0}^{y^2} f(x, y, z) \, dx \, dy \, dz \] ### **Order 5: \( dy \, dz \, dx \)** - \(x: 0 \to 6\) - For a fixed \(x\), \(y: \sqrt{x} \to 6\) - For a fixed \(y\), \(z: 0 \to 6-y\) So, \[ \int_{x=0}^6 \int_{y=\sqrt{x}}^6 \int_{z=0}^{6-y} f(x, y, z) \, dz \, dy \, dx \] ### **Order 6: \( dy \, dx \, dz \)** - \(z: 0 \to 6\) - For a fixed \(z\), \(y: 0 \to 6-z\) - For a fixed \(y\), \(x: 0 \to y^2\) So, \[ \int_{z=0}^6 \int_{y=0}^{6-z} \int_{x=0}^{y^2} f(x, y, z) \, dx \, dy \, dz \] --- ## **Fill in the Blanks:** ### **(1) \( dz \, dx \, dy \)** - \(y: 0 \to 6\) - \(x: 0 \to y^2\) - \(z: 0 \to 6-y\) ### **(2) \( dz \, dy \, dx \)** - \(x: 0 \to 6\) - \(y: \sqrt{x} \to 6\) - \(z: 0 \to 6-y\) ### **(3) \( dx \, dz \, dy \)** - \(y: 0 \to 6\) - \(z: 0 \to 6-y\) - \(x: 0 \to y^2\) ### **(4) \( dx \, dy \, dz \)** - \(z: 0 \to 6\) - \(y: 0 \to 6-z\) - \(x: 0 \to y^2\) ### **(5) \( dy \, dz \, dx \)** - \(x: 0 \to 6\) - \(y: \sqrt{x} \to 6\) - \(z: 0 \to 6-y\) ### **(6) \( dy \, dx \, dz \)** - \(z: 0 \to 6\) - \(y: 0 \to 6-z\) - \(x: 0 \to y^2\) --- ## **Final Answers in the Boxes** ### 1. \( dz \, dx \, dy \) \[ \boxed{0} \quad \boxed{6} \quad \boxed{0} \quad \boxed{y^2} \quad \boxed{0} \quad \boxed{6-y} \] ### 2. \( dz \, dy \, dx \) \[ \boxed{0} \quad \boxed{6} \quad \boxed{\sqrt{x}} \quad \boxed{6} \quad \boxed{0} \quad \boxed{6-y} \] ### 3. \( dx \, dz \, dy \) \[ \boxed{0} \quad \boxed{6} \quad \boxed{0} \quad \boxed{6-y} \quad \boxed{0} \quad \boxed{y^2} \] ### 4. \( dx \, dy \, dz \) \[ \boxed{0} \quad \boxed{6} \quad \boxed{0} \quad \boxed{6-z} \quad \boxed{0} \quad \boxed{y^2} \] ### 5. \( dy \, dz \, dx \) \[ \boxed{0} \quad \boxed{6} \quad \boxed{\sqrt{x}} \quad \boxed{6} \quad \boxed{0} \quad \boxed{6-y} \] ### 6. \( dy \, dx \, dz \) \[ \boxed{0} \quad \boxed{6} \quad \boxed{0} \quad \boxed{6-z} \quad \boxed{0} \quad \boxed{y^2} \] --- **Let me know if you want these written directly in the boxes as per the diagram!**