Problem 54 - Calculus for Biology and Medicine

Question

$$ \text { Find } \int_{-1}^{1} 3 x^{5} d x $$

Step-by-Step Solution

Answer

The definite integral \( \int_{-1}^{1} 3x^5 \, dx \) is 0.

Step 1: Identify the Integral

We are tasked with finding the definite integral \( \int_{-1}^{1} 3x^5 \, dx \). This involves integrating the function \( 3x^5 \) over the interval from \( x = -1 \) to \( x = 1 \).

Step 2: Integrate the Function

To integrate the function \( 3x^5 \), we apply the power rule of integration. The power rule states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. So, \( \int 3x^5 \, dx = 3 \cdot \frac{x^{6}}{6} = \frac{1}{2}x^6 \).

Step 3: Evaluate the Definite Integral

Since we have a definite integral, we need to evaluate the antiderivative \( \frac{1}{2}x^6 \) at the upper limit \( x = 1 \) and the lower limit \( x = -1 \), and then calculate the difference.1. Evaluate at \( x = 1 \): \( \frac{1}{2}(1)^6 = \frac{1}{2} \).2. Evaluate at \( x = -1 \): \( \frac{1}{2}(-1)^6 = \frac{1}{2} \).Subtract the evaluation at the lower limit from the evaluation at the upper limit: \[ \frac{1}{2} - \frac{1}{2} = 0 \]

Step 4: Conclusion

The result of the definite integral \( \int_{-1}^{1} 3x^5 \, dx \) is 0.

Key Concepts

Power Rule

The power rule is a fundamental technique used in calculus for finding the antiderivative or integral of a function. It simplifies the integration process, especially for polynomial functions. According to the power rule, if you have a function of the form \( x^n \), the integral is given by:\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]where \( n eq -1 \) and \( C \) represents the constant of integration. This rule is particularly helpful because it reduces the power of \( x \) by one, making the function easier to handle.

• Example: Integrating \( 3x^5 \) involves applying the power rule. The integral becomes \( 3 \cdot \frac{x^{6}}{6} \), simplifying to \( \frac{1}{2}x^6 \).
• The power rule works for any real number exponent except \( n = -1 \), as that would lead to division by zero.

By mastering the power rule, you lay a strong foundation for more advanced calculus topics.

Integration

Integration is the process of finding the integral, or the "area under the curve," of a function. Unlike differentiation, which gives us the rate of change, integration aggregates values to provide the total sum over an interval.There are two main types of integrals: indefinite and definite. Indefinite integrals, like \( \int f(x) \, dx \), include a constant \( C \) since they represent a family of functions. Definite integrals, on the other hand, have limits of integration, such as \( \int_{a}^{b} f(x) \, dx \), which calculates the net area between \( a \) and \( b \) along the x-axis.

• The function \( 3x^5 \) falls under definite integration, where the limits are \( -1 \) and \( 1 \).
• For definite integrals, evaluate the antiderivative at both limits and subtract the lower from the upper value.
• In our example, \( \frac{1}{2}(1)^6 \) minus \( \frac{1}{2}(-1)^6 \) confirms the area totals to zero.

Integration is key in various fields like engineering and physics, making it a versatile and essential tool in analysis.

Antiderivative

An antiderivative is essentially a reverse of differentiation. It is a function whose derivative results in the original function. Finding the antiderivative is the core part of integration. The antiderivative of a function \( f(x) \) is denoted by \( F(x) \) such that \( F'(x) = f(x) \).When calculating a definite integral, the antiderivative allows you to evaluate the function from one boundary to another, providing the accumulated value over that interval.

• For instance, the antiderivative of \( 3x^5 \) is \( \frac{1}{2}x^6 \), as found using the power rule.
• The definite integral \( \int_{-1}^{1} 3x^5 \, dx \) uses this antiderivative, assessed at the boundaries of \( 1 \) and \( -1 \).
• Ultimately, subtracting the two results gives the solution to the integration problem.

The concept of antiderivatives connects differential equations and integration, forming a crucial part of calculus and analytical problem-solving.