Evaluate the double integral by reversing the order of integration: ${\int }_0^4{\int }_{3y}^{12}13e^{x^2} dx dy$

• A. ${\int }_{x=0}^{x=12}{\int }_{y=0}^{y=\frac{x}{3}}13e^{x^2} dy dx$
• B. ${\int }_{x=3y}^{x=12}{\int }_{y=0}^{y=4}13e^{x^2} dy dx$
• C. ${\int }_{y=0}^{y=4}{\int }_{x=0}^{x=12}13e^{x^2} dx dy$
• D. ${\int }_{3\times 0}^{12}13e^{x^2} \left(\frac{x}{3}\right)dx$

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to reverse the order of integration for the given double integral \( \int_0^4 \int_{3y}^{12} 13e^{x^2} \, dx \, dy \). This involves analyzing the bounds of integration and rewriting them appropriately.
2. Step 2: Analyze the current bounds. The outer integral \( \int_0^4 \) corresponds to \( y \) ranging from 0 to 4. The inner integral \( \int_{3y}^{12} \) corresponds to \( x \) ranging from \( 3y \) to 12. This means \( x \) depends on \( y \).
3. Step 3: Reverse the order of integration. To do this, determine the new bounds for \( x \) and \( y \). Observe that \( x \) ranges from 0 to 12 (the original upper limit of \( x \)), and for a fixed \( x \), \( y \) ranges from 0 to \( \frac{x}{3} \) (since \( x = 3y \) implies \( y = \frac{x}{3} \)).
4. Step 4: Rewrite the integral with the reversed order of integration. The new integral becomes \( \int_{x=0}^{x=12} \int_{y=0}^{y=\frac{x}{3}} 13e^{x^2} \, dy \, dx \). This matches the correct answer provided in the problem.
5. Step 5: Verify the setup. Ensure that the new bounds correctly describe the region of integration. The region is bounded by \( y = 0 \), \( y = 4 \), \( x = 3y \), and \( x = 12 \). Reversing the order of integration correctly captures this region.