Let's analyze and solve the problem step by step. --- ### **Problem Statement Recap** You are asked to find the limits for \( x \), \( y \), and \( z \) for the solid bounded by the coordinate planes and the equation \[ z = 7 - 2x - y \] in the first octant (where \( x \geq 0, y \geq 0, z \geq 0 \)). --- ### **Step 1: Find the intercepts** #### **Intercepts with the axes:** - **x-intercept:** Set \( y = 0 \), \( z = 0 \): \[ 0 = 7 - 2x - 0 \implies x = \frac{7}{2} \] - **y-intercept:** Set \( x = 0 \), \( z = 0 \): \[ 0 = 7 - 0 - y \implies y = 7 \] - **z-intercept:** Set \( x = 0 \), \( y = 0 \): \[ z = 7 \] --- ### **Step 2: Find the inequalities** Because we are in the first octant, all variables are non-negative: \[ x \geq 0,\quad y \geq 0,\quad z \geq 0 \] Also, from the plane equation: \[ z = 7 - 2x - y \implies 7 - 2x - y \geq 0 \implies 2x + y \leq 7 \] --- ### **Step 3: Set up the bounds** Let’s express the region in the order \( dz\,dy\,dx \): - \( x \) runs from \( 0 \) to \( \frac{7}{2} \) - For each \( x \), \( y \) runs from \( 0 \) to \( 7 - 2x \) - For each \( x \) and \( y \), \( z \) runs from \( 0 \) to \( 7 - 2x - y \) --- ### **Step 4: Write the answer** **Limits:** \[ \boxed{ \begin{align*} &0 \leq x \leq \frac{7}{2} \\ &0 \leq y \leq 7 - 2x \\ &0 \leq z \leq 7 - 2x - y \\ \end{align*} } \] --- **Fill in the boxes:** - \( 0 \leq x \leq \frac{7}{2} \) - \( 0 \leq y \leq 7 - 2x \) - \( 0 \leq z \leq 7 - 2x - y \) --- **Final Answer:** \[ \boxed{ 0 \leq x \leq \frac{7}{2},\quad 0 \leq y \leq 7 - 2x,\quad 0 \leq z \leq 7 - 2x - y } \]