Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

use the Second Fundamental Theorem of Calculus to evaluate each definite integral. $$ \int_{-1}^{2} x^{4} d x $$

Step-by-Step Solution

Answer

The definite integral evaluates to \( \frac{33}{5} \).

Step 1: Identify the integrand

The function we need to integrate is the polynomial function: \( f(x) = x^4 \).

Step 2: Apply the Power Rule for Integration

To find the antiderivative of \( f(x) = x^4 \), we use the power rule: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). For \( x^4 \), it becomes \( \frac{x^5}{5} + C \).

Step 3: Evaluate the Antiderivative at the Upper Limit

Using the Second Fundamental Theorem of Calculus, evaluate the antiderivative \( \frac{x^5}{5} \) at \( x = 2 \): \[ \frac{2^5}{5} = \frac{32}{5} \]

Step 4: Evaluate the Antiderivative at the Lower Limit

Now, evaluate \( \frac{x^5}{5} \) at \( x = -1 \): \[ \frac{(-1)^5}{5} = \frac{-1}{5} \]

Step 5: Calculate the Definite Integral

Subtract the result from the lower limit from the result of the upper limit: \[ \frac{32}{5} - \left(\frac{-1}{5}\right) = \frac{32}{5} + \frac{1}{5} = \frac{33}{5} \]

Key Concepts

Definite Integral

A definite integral represents the accumulation of quantities, and it computes the net area under a curve within a given interval. In our exercise, we are dealing with the integral \[ \int_{-1}^{2} x^{4} \, dx \] which tells us to find the total area under the curve of the function \( f(x) = x^4 \) from \( x = -1 \) to \( x = 2 \).

• The process begins by finding an antiderivative, which can be thought of as the opposite of differentiation.
• After obtaining the antiderivative, we apply the Second Fundamental Theorem of Calculus, which ensures us the evaluation of the antiderivative at the upper limit minus the evaluation at the lower limit provides the value of the integral.

In simpler terms, it's about finding the total change over an interval, which is essential in many real-world applications, such as calculating distances from velocity data or determining the total growth from a rate of change.

Antiderivative

The antiderivative, or indefinite integral, is the function that reverses differentiation. Suppose we have a function \( f(x) \), its antiderivative \( F(x) \) is another function such that \( F'(x) = f(x) \). For our function \( f(x) = x^4 \), the antiderivative involves increasing the power of the term by one and then dividing by the new exponent. Given the power rule, this means:\[ \int x^4 \, dx = \frac{x^5}{5} + C \]where \( C \) represents the constant of integration, which accounts for any constant term that vanishes when deriving. In the context of definite integrals, like in our problem, the constant \( C \) cancels out when subtracting the values obtained from the upper and lower limits, which is why it's not included in the final calculation. Identifying the antiderivative correctly allows us to apply the Second Fundamental Theorem of Calculus, crucial for finding the definite integral.

Power Rule for Integration

The power rule for integration is a straightforward tool used to find the antiderivative of powers of \( x \). It is a foundational method needed to tackle polynomial functions such as those encountered often in calculus.The rule is stated as:\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]where \( n \) is not equal to \(-1\), and \( C \) is the constant of integration. Applying this rule:

• Identify \( n \) from your function. In our case, for \( f(x) = x^4 \), \( n = 4 \).
• Increase the power by one, so \( x^4 \) becomes \( x^5 \).
• Divide the new expression by the new power, yielding \( \frac{x^5}{5} \).

Thus concluding that the power rule effectively reconstructs the polynomial's original form, but now as an antiderivative, laying down the path to evaluating definite integrals efficiently.