Evaluate Integral: ∫(0 To 1) 6x⁵ Dx

In this article, we will evaluate the definite integral $\int_0^1 6x^5 dx$ using the Fundamental Theorem of Calculus. The Fundamental Theorem of Calculus provides a straightforward method for evaluating definite integrals, linking the concepts of differentiation and integration. Let's dive into the step-by-step process to solve this problem.

Understanding the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus consists of two parts. The first part states that if $F(x)$ is an antiderivative of $f(x)$, then the definite integral of $f(x)$ from $a$ to $b$ is given by:

$$\int_a^b f(x) dx = F(b) - F(a)$$

The second part states that if we define a function $F(x)$ as the integral of another function $f(t)$ from a constant $a$ to $x$, then $F'(x) = f(x)$. In our case, we will use the first part to evaluate the definite integral.

Step-by-Step Evaluation

1. Find the Antiderivative

First, we need to find the antiderivative of the function $f(x) = 6x^5$. To do this, we use the power rule for integration, which states that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $C$ is the constant of integration. Applying this rule to our function:

$$\int 6x^5 dx = 6 \int x^5 dx = 6 \cdot \frac{x^{5+1}}{5+1} + C = 6 \cdot \frac{x^6}{6} + C = x^6 + C$$

So, the antiderivative of $6x^5$ is $x^6 + C$. For the purpose of evaluating the definite integral, we can ignore the constant of integration $C$ because it will cancel out when we compute $F(b) - F(a)$.

2. Apply the Fundamental Theorem of Calculus

Now that we have the antiderivative $F(x) = x^6$, we can apply the Fundamental Theorem of Calculus to evaluate the definite integral $\int_0^1 6x^5 dx$. We need to compute $F(1) - F(0)$:

$$F(1) = (1)^6 = 1$$

$$F(0) = (0)^6 = 0$$

Therefore,

$$\int_0^1 6x^5 dx = F(1) - F(0) = 1 - 0 = 1$$

3. State the Result

The value of the definite integral $\int_0^1 6x^5 dx$ is 1.

Detailed Explanation of Each Step

Finding the Antiderivative in Detail

To find the antiderivative of $6x^5$, we use the power rule for integration. The power rule states that the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1}$, provided that $n \neq -1$. In our case, $n = 5$. Therefore, we have:

$$\int 6x^5 dx = 6 \int x^5 dx$$

We can pull the constant 6 out of the integral. Now, we apply the power rule to $\int x^5 dx$:

$$\int x^5 dx = \frac{x^{5+1}}{5+1} = \frac{x^6}{6}$$

Multiplying by the constant 6, we get:

$$6 \cdot \frac{x^6}{6} = x^6$$

Thus, the antiderivative of $6x^5$ is $x^6 + C$, where $C$ is the constant of integration. As mentioned before, we can ignore $C$ when evaluating definite integrals because it will cancel out.

Applying the Fundamental Theorem in Detail

The Fundamental Theorem of Calculus tells us that to evaluate a definite integral, we need to find the antiderivative of the function and then evaluate that antiderivative at the upper and lower limits of integration. The definite integral is the difference between the value of the antiderivative at the upper limit and the value of the antiderivative at the lower limit.

In our case, the upper limit of integration is 1, and the lower limit is 0. The antiderivative we found is $F(x) = x^6$. So, we need to compute $F(1)$ and $F(0)$:

$$F(1) = (1)^6 = 1$$

This means we substitute $x = 1$ into the antiderivative $x^6$, which gives us $1^6 = 1$.

$$F(0) = (0)^6 = 0$$

Similarly, we substitute $x = 0$ into the antiderivative $x^6$, which gives us $0^6 = 0$.

Now, we subtract the value of the antiderivative at the lower limit from the value at the upper limit:

$$\int_0^1 6x^5 dx = F(1) - F(0) = 1 - 0 = 1$$

So, the definite integral $\int_0^1 6x^5 dx$ evaluates to 1.

Practical Implications and Importance

Understanding and applying the Fundamental Theorem of Calculus is crucial in various fields such as physics, engineering, economics, and computer science. It allows us to solve problems involving rates of change, accumulation, and optimization. For example:

• Physics: Calculating the displacement of an object given its velocity function.
• Engineering: Determining the total amount of material needed for a construction project.
• Economics: Finding the consumer surplus or producer surplus in market analysis.
• Computer Science: Implementing numerical integration algorithms.

Common Mistakes to Avoid

When evaluating definite integrals, it's important to avoid common mistakes such as:

1. Forgetting the Constant of Integration: While the constant of integration $C$ cancels out in definite integrals, it's crucial to include it when finding indefinite integrals.
2. Incorrectly Applying the Power Rule: Ensure that the power rule is applied correctly, especially when dealing with negative or fractional exponents.
3. Reversing the Limits of Integration: Remember that $\int_a^b f(x) dx = - \int_b^a f(x) dx$. Reversing the limits will change the sign of the result.
4. Not Simplifying the Antiderivative: Simplifying the antiderivative before evaluating it at the limits of integration can make the calculation easier.

Conclusion

In summary, we have evaluated the definite integral $\int_0^1 6x^5 dx$ using the Fundamental Theorem of Calculus. By finding the antiderivative of $6x^5$, which is $x^6$, and then evaluating it at the limits of integration, we found that the integral equals 1. This process highlights the power and elegance of the Fundamental Theorem of Calculus in solving integration problems. Understanding these fundamental concepts allows for a deeper appreciation and application in various scientific and engineering disciplines.

For further reading and a more in-depth understanding of calculus, you can visit Khan Academy's Calculus Section.