Problem 2 - Calculus

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 value of the definite integral is \( \frac{33}{5} \).

Step 1: Identify the Antiderivative

According to the Second Fundamental Theorem of Calculus, to evaluate \( \int_{a}^{b} f(x) \, dx \), we first need to find an antiderivative \( F(x) \) of \( f(x) = x^4 \). The antiderivative of \( x^4 \) is \( F(x) = \frac{x^5}{5} \).

Step 2: Apply the Antiderivative to the Limits

After finding the antiderivative, substitute the upper and lower limits into the antiderivative. Here, we calculate \( F(2) \) and \( F(-1) \):- \( F(2) = \frac{2^5}{5} = \frac{32}{5} \)- \( F(-1) = \frac{(-1)^5}{5} = -\frac{1}{5} \)

Step 3: Compute the Definite Integral

Subtract \( F(-1) \) from \( F(2) \) to find the value of the definite integral:\[ \int_{-1}^{2} x^{4} \, dx = F(2) - F(-1) = \frac{32}{5} - \left(-\frac{1}{5}\right) \]Simplify this to get:\[ \frac{32}{5} + \frac{1}{5} = \frac{33}{5} \]

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus that represents the signed area under a curve defined by a function, over a specific interval from \( a \) to \( b \). This is different from an indefinite integral, which refers to the general antiderivative of a function without specific bounds.
When solving a definite integral like \( \int_{-1}^{2} x^{4} \, dx \), you're essentially finding the net area between the curve and the x-axis, from \( x = -1 \) to \( x = 2 \).
This calculation involves:

• Finding the Antiderivative: Determine the antiderivative (or original function) of the integrand.
• Evaluating the Antiderivative: Substitute the upper and lower limits into this antiderivative.
• Subtracting Values: Subtract the value of the antiderivative at the lower limit from its value at the upper limit.

This gives you the result of the definite integral.

Antiderivative

The antiderivative, or primitive, of a function is a function whose derivative is the original function. To find an antiderivative, you're essentially reversing the process of differentiation.
In the exercise, to solve \( \int_{-1}^{2} x^{4} \), we look for a function \( F(x) \) such that \( F'(x) = x^4 \).
The antiderivative of \( x^4 \) turns out to be \( F(x) = \frac{x^5}{5} \).

• This antiderivative is found using integration rules that reverse the power rule of differentiation.
• It's crucial to ensure that the antiderivative is correctly derived, which can be confirmed by differentiating \( F(x) \) and checking it equals the original function \( x^4 \).

With the antiderivative, you can proceed to solve the definite integral by evaluation and subtraction.

Power Rule

The power rule is a basic tool in calculus used for differentiation and integration. For integration, it helps find antiderivatives of power functions.
For any function \( x^n \), the power rule for integration is expressed as:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)
• Here, \( n \) is any real number except \( -1 \), and \( C \) is the constant of integration for indefinite integrals.

In the given problem \( \int_{-1}^{2} x^{4} \, dx \), the power rule is used to find the antiderivative:

• Increases the exponent \( 4 \) by one to get \( 5 \).
• Divides by the new exponent, giving the antiderivative \( \frac{x^5}{5} \).

The power rule simplifies the process of integration and is foundational for solving many calculus problems. Understanding this rule allows one to handle a wide variety of integrals with ease.