Let's go through the steps: ### 1. Analyze the Integral Given: \[ \int_0^9 \int_0^y f(x, y)\, dx\, dy \] - **Outer integral:** \( y \) goes from \( 0 \) to \( 9 \). - **Inner integral:** \( x \) goes from \( 0 \) to \( y \). This means \( x \) starts at 0 and goes up to \( y \), and \( y \) goes from 0 to 9. ### 2. Sketch the Region - \( 0 \leq x \leq y \) - \( 0 \leq y \leq 9 \) This region is **below the line \( y = x \), above \( x = 0 \), and between \( y = 0 \) and \( y = 9 \)**. #### Let's check the options in the picture: - The **top-left graph** is shaded below \( y = x \), between \( x = 0 \) and \( y = 9 \). This is the correct region. - The other graphs do not match this region. **So, the correct answer is the top-left graph, which is already selected.** --- ### 3. Change the Order of Integration The original region was: - \( 0 \leq x \leq y \leq 9 \) To change the order, solve for \( y \) in terms of \( x \): - For a fixed \( x \), \( y \) goes from \( x \) (on the line \( y = x \)) up to \( 9 \). - \( x \) goes from 0 to 9. So, the new limits are: - \( x \) from 0 to 9 - \( y \) from \( x \) to 9 Thus, the answer is: \[ \int_0^9 \int_x^9 f(x, y)\, dy\, dx \] **Fill in the blanks for the order-changed integral:** \[ \int_0^9 \int_x^9 f(x, y)\, dy\, dx \] --- ## **Summary** ### - The correct region is the **top-left graph**. ### - The new order of integration is: \[ \boxed{ \int_0^9 \int_x^9 f(x, y)\, dy\, dx } \]

Let's analyze the problem step-by-step, including detailed reasoning for each part, including the options and the final answer. --- ## Step 1: Understand the original integral and the region Given integral: \[ \int_{y=0}^{9} \int_{x=0}^{y} f(x, y) \, dx \, dy \] **Limits:** - Inner integral: \(x\) from 0 to \(y\) - Outer integral: \(y\) from 0 to 9 **Region in the xy-plane:** - For each fixed \( y \), \( x \) varies from 0 to \( y \). - \( y \) varies from 0 to 9. **Geometric interpretation:** - The region is bounded below by \( y=0 \), - above by \( y=9 \), - and on the left by \( x=0 \), - and on the right by \( x=y \). This describes the region **below the line \( y = x \)**, between \( y=0 \) and \( y=9 \). --- ## Step 2: Sketch the region (based on the options) The question provides multiple graphs, and the goal is to identify which one correctly depicts the region. **Analyzing options:** - **Top-left graph:** - The shaded region is **bounded below by \( y=0 \)**, - **left boundary at \( x=0 \)**, - **upper boundary at \( y=9 \)**, - **diagonal boundary along \( y=x \)**. This matches the description of the region where \( 0 \leq x \leq y \leq 9 \). - **Other graphs:** - Some may show regions above \( y=x \) or bounded differently. - Since the original limits are \( x \leq y \), the correct region is **below \( y=x \)**. **Conclusion:** - The **top-left graph** correctly depicts the region described by the limits. --- ## Step 3: Change the order of integration Original limits: \[ x: 0 \to y \\ y: 0 \to 9 \] **To change the order:** - For a fixed \( x \), what are the limits of \( y \)? Since \( x \leq y \leq 9 \), for each \( x \), \( y \) varies from \( y=x \) up to \( y=9 \). - \( x \) varies from 0 to 9, covering the full projection of the region on the x-axis. **New limits:** \[ x: 0 \to 9 \\ y: x \to 9 \] **New integral:** \[ \boxed{ \int_{x=0}^9 \int_{y=x}^9 f(x, y) \, dy \, dx } \] --- ## Step 4: Summarize each option with detailed reasoning ### **Option 1 (original integral):** \[ \int_0^9 \int_0^y f(x, y) \, dx \, dy \] - **Limits:** \( y \) from 0 to 9, \( x \) from 0 to \( y \). - **Region:** **below the line \( y=x \)**, between \( y=0 \) and \( y=9 \). - **The graph:** matches the top-left sketch. --- ### **Option 2 (change of order):** \[ \int_0^9 \int_x^9 f(x, y) \, dy \, dx \] - **Limits:** \( x \) from 0 to 9, \( y \) from \( x \) to 9. - **Region:** **above the line \( y=x \)**, between \( x=0 \) and \( x=9 \). - **Corresponds to:** the same region, but with the order of integration swapped. --- ### **Option 3 and others:** - These may depict incorrect regions or incorrect limits based on the initial description. **Key points:** - The region **is below \( y=x \)**, between \( y=0 \) and \( y=9 \). - The **correct change of limits** involves \( x \) from 0 to 9, and for each \( x \), \( y \) from \( x \) to 9. --- ## **Final answer:** ### **The correct sketch:** - The **top-left graph** accurately depicts the region. ### **The change of the order of integration:** \[ \boxed{ \int_0^9 \int_x^9 f(x, y) \, dy \, dx } \] --- ## **Summary in conclusion:** - **Region:** The set of points where \( 0 \leq x \leq y \leq 9 \), i.e., **below the line \( y=x \)**, between \( y=0 \) and \( y=9 \). - **Sketch:** Top-left graph. - **Order of integration after change:** \[ \boxed{ \int_0^9 \int_x^9 f(x, y) \, dy \, dx } \] This process ensures a clear understanding of the region and proper limits for integration.