Question

Prove that $$\int_{a}^{b} x^{2} d x=\frac{b^{3}-a^{3}}{3}$$

Short answer

The antiderivative (indefinite integral) of \( x^{2} \) over the interval \( [a, b] \) is \( \frac{b^3 - a^3}{3} \), as proven by the Fundamental theorem of Calculus.

Step-by-step solution

Indefinite Integral of \( x^{2} \)

Start by integrating \( x^2 \) with respect to \( x \) to find the antiderivative. The integral of \( x^2 \) is \( \frac{1}{3}x^3 + C \) where \( C \) is the constant of integration. We will ignore \( C \) for now as it will cancel out when we apply the limits of integration.

Application of the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if a function is continuous over the interval [a, b] then the definite integral of the function from a to b is equal to the antiderivative evaluated at \( b \) minus the antiderivative evaluated at \( a \). So, let's apply the theorem here. We end up with \( [\frac{1}{3}b^3] - [\frac{1}{3}a^3] \)

Simplify the Result

Simplifying the result from step 2, we get \( \frac{b^3 - a^3}{3} \). Thus, proving that \( \int_{a}^{b} x^{2} dx = \frac{b^3 - a^3}{3} \)

Key concepts

Antiderivative Calculation

Understanding the calculation of an antiderivative, often referred to as taking the indefinite integral, is an essential concept in calculus. To find an antiderivative means to determine the original function before differentiation. Let's use the given exercise to illustrate this concept.

When we take the integral of a function such as \( x^2 \), we are essentially asking: what function could I differentiate to get \( x^2 \) back as the derivative? If we apply the power rule in reverse, we increment the exponent by one and divide by the new exponent. Doing this for \( x^2 \), we get the antiderivative \( \frac{1}{3}x^3 \), signifying that \( \frac{d}{dx}(\frac{1}{3}x^3) = x^2 \). The constant of integration, \( C \), a byproduct of the antiderivative process, accounts for any constant that could have been present originally since the derivative of a constant is zero.

In the context of definite integrals, the constant \( C \) cancels out when evaluating the limits of integration, which is why it can be momentarily set aside during the calculation.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a critical link between the concepts of differentiation and integration in calculus. It provides a powerful tool for evaluating definite integrals. It bridges the gap between anti-differentiation and the accumulated area under a curve.

This theorem states that if a function \( f \) is continuous over the interval \( [a, b] \), then the definite integral of \( f \) from \( a \) to \( b \) can be obtained by taking the difference between its antiderivative evaluated at the upper limit, \( b \), and the antiderivative evaluated at the lower limit, \( a \). Symbolically, \( \int_a^b f(x) dx = F(b) - F(a) \), where \( F \) is any antiderivative of \( f \).

Applying this theorem to our problem, we previously found that the antiderivative of \( x^2 \) is \( \frac{1}{3}x^3 \). Using the theorem, we evaluate the antiderivative at the upper and lower limits of the integral and subtract the latter from the former, resulting in \( \frac{b^3 - a^3}{3} \). This concludes the accurate evaluation of the definite integral using one of the most profound principles in calculus.

Limits of Integration

The limits of integration are the boundaries that define the region over which we are integrating. They are denoted as the lower limit \( a \) and the upper limit \( b \) in the definite integral \( \int_a^b f(x) dx \). These limits instruct us on where to start and stop the process of accumulating area under the curve of the function.

For our exercise, the limits given were \( a \) and \( b \). When we substitute these values into our antiderivative, \( \frac{1}{3}x^3 \), we perform two separate calculations: once for \( x = b \) and once for \( x = a \), and then we find the difference between them. This step effectively captures the total