91Ó°ÊÓ

An antiderivative, also called an indefinite integral, is a function whose derivative is the original function given. For instance, if you have a function \( f(x) \), finding an antiderivative means finding a function \( F(x) \) such that \( F'(x) = f(x) \). In this exercise, you need to find the antiderivative of \( x^2 \). When working with polynomials, the process can feel repetitive but remember, each term in your polynomial is addressed individually using known rules.

The power rule is a fundamental tool in calculus for finding antiderivatives. It states that for any real number \( n \) other than \( -1 \), the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. Applying this rule to our example, \( x^2 \), you increment the power of \( x \) by 1 to get \( x^3 \), and then divide by the new power (3). Therefore, the antiderivative of \( x^2 \) is \( \frac{x^3}{3} + C \). This simple but powerful rule is essential for solving many integral problems efficiently.

Evaluating a definite integral means finding the value of the antiderivative at the upper and lower limits of integration and then taking the difference. In our exercise, the definite integral of \( x^2 \) from 0 to 2 requires you to plug in these limits into the antiderivative. Here’s the step-by-step:

1. Find the antiderivative: \( \frac{x^3}{3} \).
2. Substitute the upper limit (2): \( \frac{2^3}{3} = \frac{8}{3} \).
3. Substitute the lower limit (0): \( \frac{0^3}{3} = 0 \).
4. Subtract the lower limit result from the upper limit result: \( \frac{8}{3} - 0 = \frac{8}{3} \).

This stepwise evaluation gives us the area under the curve \( x^2 \) from 0 to 2, which is \( \frac{8}{3} \). Knowing how to evaluate these limits correctly ensures accurate results in your definite integrals.