91Ó°ÊÓ

Integral calculus is a branch of mathematics that deals with the accumulation of quantities. It involves calculating the total size, length, area, or other cumulative aspects over a continuous interval. In the context of probability density functions, integral calculus helps in finding total probabilities over an interval. This is essential for confirming that a function can serve as a valid probability distribution.

When verifying the properties of a probability density function (PDF), we often compute the integral of the function over its given interval. This process involves:

• Identifying the function to integrate, which in this case, is the constant function \(f(x) = \frac{1}{5}\).
• Specifying the interval over which to integrate, [3, 8].
• Evaluating the integral in order to verify it sums to 1, thus validating the PDF.

This use of integral calculus ensures the entire area under the function's curve on a specified interval equates to 1, as it denotes the whole probability sum.

Antiderivatives, also known as indefinite integrals, are the reverse operation of derivatives. When we have a function whose derivative we know, the antiderivative helps us go back to the original function.

For example, in the provided exercise, the function \( f(x) = \frac{1}{5} \) is a constant function. The antiderivative of a constant \( c \) is given by \( cx + C \), where \( C \) is a constant. Therefore, the antiderivative of \( \frac{1}{5} \) is \( \frac{1}{5}x + C \).

When verifying a probability density function, the antiderivative is critical as it is required to calculate definite integrals over a given interval. Here, it helps transform our integral into a form where limits can be easily applied to extract a numerical probability value.

The limits of integration define the interval over which we evaluate a definite integral. In probability density functions, these limits correspond to the endpoints of the interval for which the function is defined to calculate total probability.

For the exercise, the limits of integration are [3, 8]. After finding the antiderivative \( \frac{1}{5}x \), the next step is to substitute these limits into the antiderivative:

• First, substitute the upper limit into the antiderivative: \( \frac{1}{5}(8) \).
• Next, substitute the lower limit: \( \frac{1}{5}(3) \).
• Subtract the value at the lower limit from the value at the upper limit to get the net area under the curve.

This process delivers the definite integral result, ensuring the area under the function's graph over the interval equates to 1, confirming valid calculus and probability assertions. Thus, using limits of integration verifies both mathematical accuracy and conceptual validity within probability theory.