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A definite integral calculates the net area under a curve between two specified bounds, which are typically denoted as the lower and upper limits of integration. In simple terms, it provides the accumulation or total of quantities over a given interval. For example, in the iterated integral \[ \int_{0}^{\ln 3} \int_{0}^{\ln 2} e^{x+y} \, dy \, dx \]we are finding the total accumulation of the function \( e^{x+y} \) over the region defined by the given limits. When evaluating a definite integral, we first need to find the antiderivative of the function with respect to one variable, and then evaluate it using the given limits. Here's the step-by-step process:

• Integrate: Find the antiderivative of the function with respect to the inner variable, which in this case is \( y \).
• Evaluate: Substitute the limits of the inner integral and subtract to find the total accumulation along that dimension.
• Repeat: Use the result to perform the integration with respect to the next variable \( x \).

This process helps us understand how a function behaves over an entire region, rather than just at one specific point.

Exponential functions are mathematical expressions that contain a constant base raised to a variable exponent, typically of the form \( f(x) = a^x \) or \( e^x \). In calculus, the natural exponential function \( e^x \) is especially significant due to its unique properties, including its simple derivative and integral. In our exercise, the function \( e^{x+y} \) is a product of exponential terms involving two variables. This feature allows us to separate and simplify expressions during integration, as illustrated below:

• Multiplicative Identity: Exponential terms can be broken apart using properties such as \( e^{x+y} = e^x \cdot e^y \).
• Simplification: This property enables us to simplify \( e^{x + \ln 2} \) as \( 2e^x \). The expression \( e^{\ln 2} = 2 \) since \( e^{\ln a} = a \).
• Calculus Operations: The derivative and integral of \( e^x \) are both \( e^x \), making it straightforward to integrate functions involving \( e^x \).

Exponential functions appear in various fields including compound interest calculations, population growth models, and radioactive decay, making them crucial for mathematical modeling.

The antiderivative, often called the indefinite integral, is the reverse operation of differentiation. It is a function that, when differentiated, gives the original function back. For instance, if \( F(x) \) is an antiderivative of \( f(x) \), then \( F'(x) = f(x) \). In the context of the iterated integral problem, we need to find antiderivatives to solve definite integrals.### Antiderivative of \( e^{x+y} \) with respect to \( y \)Here, the antiderivative of \( e^{x+y} \) with respect to \( y \) is still \( e^{x+y} \), as differentiation of \( e^{x+y} \) with respect to \( y \) would simply reproduce the same function. Here's why:

• When taking the integral with respect to \( y \), consider \( x \) as a constant, thus the integral operates just as \( e^{y} \).
• After integration, applying the limits of \( y = 0 \) to \( y = \ln 2 \) results in the expression \( e^{x + \ln 2} - e^{x} \).

The next step involves simplifying terms and finding the antiderivative with respect to \( x \), leading us to the final evaluation.Recognizing and applying antiderivatives is critical because they allow you to transition from the original function to its total or accumulated change over an interval.