Question

In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { - 1 } ^ { 2 } 3 x ^ { 2 } d x$$

Short answer

The result of the integral \(\int_{-1}^{2} 3x^2 dx\) is 9.

Step-by-step solution

Identify the function and limits of integration

The function to be integrated is \(3x^2\), and the limits of integration are -1 (lower limit) and 2 (upper limit).

Find the antiderivative of the function

The antiderivative of \(3x^2\) can be found by using the power rule for integration, which says the integral of \(x^n\) is \(\frac{1}{n+1}x^{n+1}\). In this case, n is 2, so the antiderivative of \(3x^2\) is \(\frac{3}{2+1}x^{2+1}\) which simplifies to \(x^3\).

Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that the definite integral of a function from a to b is equal to the antiderivative at b minus the antiderivative at a. So substitute the upper limit (2) into the antiderivative function to find its value at this point, and then do the same for the lower limit (-1). Thus, it becomes \([2^3 - (-1)^3]\), which equates to \([8 - (-1)]\). Following the arithmetic operations, the result is 9.

Key concepts

Antiderivatives

When working with calculus, one of the central tasks is to find the original function given its derivative—this is known as finding the antiderivative or indefinite integral. The process of integration is essentially the reverse of differentiation. To find antiderivatives, you typically apply integration rules that correspond to the differentiation rules you might already be familiar with. For example, if you know that the derivative of \(x^n\) is \(nx^{n-1}\), then by reversing this process, you can conclude that an antiderivative of \(ax^{n}\) would be \(\frac{a}{n+1}x^{n+1}\) plus a constant of integration, usually represented by \(C\).

In solving the integral \(\int _ { - 1 } ^ { 2 } 3 x ^ { 2 } d x\), we look for the antiderivative of \(3x^2\). By applying the power rule, we determine that the antiderivative, in this case, is \(x^3\), without the need for the constant \(C\) because we are dealing with a definite integral.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a critical link between the fields of differentiation and integration. It has two parts; the first part provides an easy way to find an antiderivative of a function, while the second part (often the focus in the application of the definite integral) relates the definite integral of a function to its antiderivative. To be more specific, the second part states that if \(F\) is a continuous antiderivative of a function \(f\) on an interval \([a, b]\), then \(\int_{a}^{b} f(x) dx = F(b) - F(a)\).

By using this theorem on our example, we evaluated the antiderivative of \(3x^2\), which is \(x^3\), at the limits of integration, subtracting the antiderivative evaluated at the lower limit \(-1\) from that evaluated at the upper limit \(2\), giving us the result of 9.

Power Rule for Integration

Integration can sometimes be approached with patterns and rules, one of which is the power rule for integration. This rule is an essential tool for finding the antiderivative of polynomials. The power rule states that the integral of \(x^n\), for any real number \(n\) not equal to -1, is given by \(\frac{x^{n+1}}{n+1}\) plus a constant of integration \(C\). The power rule is a direct result of reversing the differentiation process of power functions.

In our exercise, applying the power rule involves integrating the function \(3x^2\). Following the rule, we increment the exponent by one (yielding 3) and divide by this new exponent, therefore obtaining the antiderivative \(x^3\), as shown in the solution.

Limits of Integration

Limits of integration specify the interval over which we integrate a given function. They define the starting and ending points—also known as the lower and upper limits—for the process of integration. In a definite integral, these limits are the boundaries between which we sum up the infinitesimal elements defined by the function being integrated. The result is the net area under the curve of the function between these two points.

For the given problem, the limits of integration are -1 and 2. When we apply the Fundamental Theorem of Calculus, we use these limits to evaluate the antiderivative at both the lower and upper points and subtract the former from the latter. Hence, the limits of integration are crucial in providing the exact numeric value of a definite integral.