a) \( \int_{0}^{1} \int_{0}^{2} y e^{x y} d x d y=\frac{e^{2}-2}{2} \)
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Let's explore the marvels of double integrals! This particular integral evaluates the area under the curve defined by the function \( y e^{xy} \) over the rectangle defined by \( 0 \leq x \leq 2 \) and \( 0 \leq y \leq 1 \). It highlights the beauty of integrating functions that intersect with exponential growth, showcasing how these functions can illustrate real-world phenomena such as population growth or the spread of information, translating mathematical elegance into tangible scenarios. Now, about solving double integrals! It's crucial to carefully sketch the region of integration to avoid miscalculations. A common mistake happens when switching the order of integration without appropriately adjusting the limits. Ensure you’re properly interpreting the bounds when changing the order; otherwise, your results might be as far off as a misplaced decimal! Remember, every integral tells a story, and the bounds help narrate that tale accurately!