91Ó°ÊÓ

In calculus, integration is a fundamental concept that helps us find areas under curves. When dealing with probability density functions (PDFs), integration is used to ensure the total area under the curve meets certain conditions.
For a function to be a valid PDF over an interval \([a, b]\), the integral of the function over this interval must equal 1. This means we need to find \[ \int_{a}^{b} f(x) \, dx = 1 \] to confirm the function is a proper PDF.
This process essentially checks that the total probability across the interval is 100%. It’s a crucial part of probability and statistics to ensure models are accurate and reliable.

To solve an integral, you need to follow a systematic approach. Let’s break down the steps using the function provided in the exercise.
First, set up the integral of the function over the given interval. In this exercise, the function is \(f(x) = kx^2\) and the interval is \[0 \leq x \leq 2 \]. This means we need to solve: \[\begin{aligned} \int_{0}^{2} kx^2 \, dx \end{aligned}\].
Next, we integrate the function. For \(x^2\), the antiderivative is \(\frac{x^3}{3}\) because the power rule states that the integral of \(x^n\) is \(\frac{x^{(n+1)}}{n+1}\) plus a constant. Applying this, we get: \[ \int_{0}^{2} x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^2 \].
Step-by-step, this method ensures accuracy and consistency in solving calculus problems.

Calculus problems, specifically those involving integration, are pervasive in studying probability density functions. In our exercise, the problem requires calculating the integral of a quadratic function multiplied by a constant over a specified interval.
This focuses on understanding how different functions behave over certain ranges and ensuring our probabilities sum to 1. By integrating the function and balancing it equal to 1, we solve for our constant, making our function a valid PDF. This involves:
Setting up the integral
Solving the integral
Setting the result equal to 1
Solving for any constants involved, like \(k\) in this exercise . Such problems deepen our understanding of how mathematical concepts apply in real-world scenarios.
They also highlight the importance of precise calculations and verification steps to maintain the integrity of mathematical models.

A definite integral is evaluated over a specific interval and has numerous applications in probability. In our exercise, we used a definite integral to confirm that the function \(f(x) = kx^2\) is a PDF on the interval \[0, 2 \].
The definite integral in this context is \[ \int_{0}^{2} kx^2 \ dx = k \frac{8}{3} \]. This integral represents the area under the curve from \(x = 0\) to \(x = 2\).
By setting this equal to 1 (which is a must for PDFs), we solve for \(k\). In this case, \(\frac{8}{3}k = 1\) which simplifies to \(k = \frac{3}{8}\). This constant ensures the function properly normalizes across the specified interval, validating it as a probability density function.
Understanding definite integrals bridges abstract mathematical concepts with practical applications in probability and statistics, making it an essential tool for students.