Answer
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Definite integrals and area are intimately related in calculus. The definite integral of a function \( f(x) \) over a closed interval \([a, b]\) represents the signed area between the graph of the function and the x-axis over that interval.
Mathematically, if \( f(x) \) is a continuous function defined on \([a, b]\), then the definite integral of \( f(x) \) from \( a \) to \( b \) is denoted by:
\[ \int_{a}^{b} f(x) \, dx \]
Geometrically, this integral represents the area bounded by the curve \( f(x) \), the x-axis, and the vertical lines \( x = a \) and \( x = b \). The integral accounts for areas above the x-axis as positive and areas below the x-axis as negative.
Other interpretations of definite integrals include:
1. Accumulation of Quantity: The definite integral can represent the accumulation of a quantity over a certain interval. For example, if \( f(x) \) represents the rate of change of a quantity with respect to \( x \), then the definite integral of \( f(x) \) over an interval \([a, b]\) gives the total change of the quantity over that interval.
2. Average Value of a Function: The definite integral divided by the width of the interval can give the average value of the function \( f(x) \) over that interval.
3. Length of a Curve: In some cases, definite integrals can represent the length of a curve. For instance, if \( f(x) \) represents a curve in the plane, then the integral of \( \sqrt{1 + (f'(x))^2} \) over an interval \([a, b]\) gives the length of the curve corresponding to \( f(x) \) over that interval.
4. Probability Density Functions: In probability theory, definite integrals are used to calculate probabilities. The integral of a probability density function over a certain interval gives the probability of the random variable falling within that interval.
These interpretations highlight the versatility of definite integrals beyond just measuring area and provide insight into various applications across mathematics and other fields.
