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Calculating the area under a curve is a fundamental concept in integral calculus. It involves finding the integral of a function over a specific interval. The integral represents the accumulated quantity, and in the context of the area under a curve, it quantifies the space between the graph of the function and the x-axis over that interval.
To find the area under the curve of a function like \(y = e^x\) over the interval \([-1, 5]\), we use the definite integral:
\[ \text{Area} = \int_{-1}^{5} e^{x} \, dx \] This integral will give us the total area by summing up infinitesimal areas represented by the rectangular strips with height \(e^x\) and infinitely small width \(dx\). This concept is useful in many practical applications like physics and engineering, where it helps in determining quantities that are accumulated over a range.
Examples include finding distances, areas, and volumes.

The antiderivative, also known as the indefinite integral, is the reverse process of differentiation. Given a function, the antiderivative is a function whose derivative is the original function. For instance, if we are given the function \(e^x\), we know that:
\[ \frac{d}{dx} e^x = e^x \] Hence, the antiderivative of \(e^x\) is just \(e^x\).
In the problem, we compute the antiderivative to evaluate the definite integral:
\[ \int e^x \, dx = e^x + C \] Where \(C\) is the constant of integration. However, when dealing with definite integrals, the constant \(C\) cancels out, simplifying our computation.

The exponential function, \(e^x\), is one of the most important functions in mathematics and appears frequently in many areas of science and engineering. It is defined as a function where the base is the mathematical constant \(e\) (approximately equal to 2.71828) and the exponent is the variable \(x\).
Key properties of the exponential function include:

• The derivative of \(e^x\) with respect to \(x\) is \(e^x\).
• The function \(e^x\) is always positive.
• \(e^x\) grows faster than any polynomial function.

In the exercise, we utilized the exponential function \(e^x\) to demonstrate how to calculate the area under its curve over a specified interval \([-1,5]\). The special property of \(e^x\) being its own derivative simplifies finding its antiderivative, making it straightforward to evaluate the associated definite integral.