91Ó°ÊÓ

To evaluate a definite integral, one crucial concept is the antiderivative. An antiderivative of a function is another function that achieves the role of reversing differentiation. This means, when you differentiate the antiderivative, you return to the original function. The antiderivative is integral in calculating the overall area under a curve over a specified interval. Often, it is denoted by the integral symbol without the boundaries of integration. Additionally, this concept forms the basis for finding solutions to many calculus problems. In our exercise, the given integrand is \( x^{1/3} \). To find its antiderivative, observe what happens when you apply certain rules of integration.

The power rule for integration is a fundamental tool when finding the antiderivative of functions with variables raised to a power. It states that if you have an integrand of the form \( x^n \), where \( n eq -1 \), the antiderivative is \( \frac{x^{n+1}}{n+1} \). This means you add 1 to the exponent and divide by this new exponent. In our case, the integrand is \( x^{1/3} \). Applying the power rule:

• Increase the exponent by 1, resulting in \( x^{4/3} \).
• Divide by the new exponent, resulting in \( \frac{3}{4} x^{4/3} \).

This gives us the antiderivative \( \frac{3}{4} x^{4/3} \). This straightforward tool simplifies the process of finding antiderivatives when dealing with powers of \( x \).

The Fundamental Theorem of Calculus beautifully links differentiation and integration. This theorem comes in two parts, with the second part being particularly useful for evaluating definite integrals. It states that if you have a continuous function over [a, b] and \( F \) as its antiderivative, the definite integral is calculated as: \[ \int_a^b f(x) \, dx = F(b) - F(a) \] To apply this, find the antiderivative, evaluate it at the upper limit b, and subtract the evaluation at the lower limit a. In our exercise, the antiderivative of \( x^{1/3} \) is \( \frac{3}{4} x^{4/3} \). Thus the calculation goes:

• Evaluate \( \frac{3}{4} \times 27^{4/3} - \frac{3}{4} \times 8^{4/3} \).
• With previous calculations, this results in 81 and 16, respectively.
• Subtract to get \( \frac{195}{4} \).

This theorem is fundamental as it allows the integration process to be completed by mere evaluation and subtraction.