Problem 29 - Calculus Early Transcendentals

Question

Evaluate the following integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{2} 4 x^{3} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the definite integral $$\int_{0}^{2} 4 x^{3} d x$$ using the Fundamental Theorem of Calculus. Answer: The definite integral evaluates to 16.

Step 1: Find the Antiderivative of the Given Function

We need to find the antiderivative of the function 4x^3. Using the power rule of integration (integral of $$x^n = \frac{x^{n+1}}{n+1} + C$$), we can find the antiderivative: $$\int_{}^{} 4 x^{3} d x = \frac{4 x^{4}}{4} + C$$ Simplifying the result: $$\int_{}^{} 4 x^{3} d x = x^{4} + C$$

Step 2: Apply the Fundamental Theorem of Calculus

Using the Fundamental Theorem of Calculus, we can evaluate the definite integral by finding the difference of the antiderivative evaluated at the endpoints. In our case, the endpoints are 0 and 2. So, we evaluate the antiderivative at these points: $$ F(2) = (2^{4} + C) = 16 + C$$ $$ F(0) = (0^{4} + C) = 0 + C$$ Now compute F(2) - F(0): $$\int_{0}^{2} 4 x^{3} d x = F(2) - F(0) = (16 + C) - (0 + C) = 16$$ So, the value of the definite integral is 16.

Key Concepts

Definite Integral

The concept of a definite integral is fundamental in calculus, encapsulating the idea of the net area under a curve over a certain interval. To visualize this, imagine plotting a curve on a graph and looking at the space between the curve and the x-axis from one point to another. The definite integral, symbolized using the integral sign with upper and lower limits, quantifies this area with a numerical value.

When you're asked to evaluate something like \( \int_{0}^{2} 4 x^{3} d x \) what you're essentially doing is finding the total accumulated value of \(4x^3\) as x moves from 0 to 2. To achieve this, you need to find the values at the boundary points and subtract the lower bound value from the upper bound value. One of the key aspects here is to recognize that the definite integral results in a number, not another function or equation.

The solution given shows the definite integral being evaluated by applying the Fundamental Theorem of Calculus. Here, we use the theorem to transform our problem into a much simpler one - instead of finding areas directly, we find the difference between two evaluations of the antiderivative function.

Antiderivative

In calculus, an antiderivative is a function that reverses the process of differentiation. In essence, finding an antiderivative means identifying a function which, when differentiated, yields the function you started with. For example, if you have \(4x^3\), an antiderivative of this function would be a function whose derivative gives back \(4x^3\).

The antiderivative is not unique, as it can differ by a constant—hence the \(+ C\) in the result, known as the constant of integration. This represents an infinite family of functions, all of which are valid antiderivatives. When computing a definite integral, the constant of integration cancels out, as shown in the step-by-step solution, leaving a straightforward numeric value representing the area under the curve.

Power Rule of Integration

The power rule of integration is a quick method to find the antiderivative of a function in the form of \( x^n \), where \( n \) is any real number except -1. The rule states that the integral of \(x^n\) with respect to \(x\) is \(\frac{x^{n+1}}{n+1}\), plus the constant of integration. This rule automates the process of integration when dealing with polynomial functions.

For instance, in our exercise, the power rule is applied to, \( \int 4x^3 dx \), which simplifies to \( \frac{4 x^{4}}{4} + C \) and then further to \( x^{4} + C \). It's essential to note the \(+ C\), even though it will not affect the definite integral's evaluation. The power rule of integration is an indispensable tool for finding antiderivatives, especially when dealing with definite integrals as it simplifies the step where you subtract the boundary values to find the area under a curve.