What Is The Value Of The Definite Integral $\int_1^a [-1 + H(x)]dx$, Assuming $h(x) = X^2$?

In this article, we will delve into the concept of definite integrals and explore how to solve them. We will use a specific example to illustrate the process, providing a step-by-step guide for better understanding. Our example involves the function $f(x) = x^2$ and the integral $\int_1^a [-1 + h(x)]dx$. We will calculate the value of this definite integral, breaking down each step to ensure clarity.

Introduction to Definite Integrals

Before diving into the specifics of our problem, let's establish a foundational understanding of what definite integrals are. In calculus, a definite integral is a way to calculate the area under a curve between two specified limits. Imagine a function plotted on a graph; the definite integral gives you the area bounded by the function's curve, the x-axis, and the vertical lines at the limits of integration. This concept is widely used in physics, engineering, economics, and various other fields to model and solve real-world problems.

The definite integral is represented mathematically as:

$$\int_a^b f(x) dx$$

Here, $f(x)$ is the function being integrated, $a$ is the lower limit of integration, $b$ is the upper limit of integration, and $dx$ indicates that we are integrating with respect to $x$. The result of a definite integral is a numerical value, unlike an indefinite integral which results in a family of functions.

Problem Statement

Now, let's focus on the problem at hand. We are given the function $f(x) = x^2$ and the definite integral:

$$\int_1^a [-1 + h(x)]dx$$

We need to determine the value of this integral. However, there seems to be a slight ambiguity in the problem statement. The function $h(x)$ is not explicitly defined. To proceed, we need to make an assumption about $h(x)$. A reasonable assumption, given the context of the function $f(x) = x^2$, is that $h(x)$ might be related to $f(x)$. Let's assume that $h(x) = f(x) = x^2$. This assumption will allow us to solve the integral step by step. If this assumption is incorrect, the problem would require additional information to be solved.

With the assumption that $h(x) = x^2$, our integral becomes:

$$\int_1^a [-1 + x^2]dx$$

The problem now involves finding the definite integral of the function $-1 + x^2$ from $1$ to $a$. This is a straightforward calculus problem that we can solve using the power rule for integration and the fundamental theorem of calculus.

Step-by-Step Solution

Step 1: Find the Indefinite Integral

The first step in evaluating a definite integral is to find the indefinite integral of the function. The indefinite integral is the antiderivative of the function. For our function $-1 + x^2$, we need to find a function whose derivative is $-1 + x^2$. We can 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:

$$\int (-1 + x^2) dx = \int -1 dx + \int x^2 dx$$

$$\int -1 dx = -x + C_1$$

$$\int x^2 dx = \frac{x^3}{3} + C_2$$

Combining these results, we get the indefinite integral:

$$\int (-1 + x^2) dx = -x + \frac{x^3}{3} + C$$

where $C = C_1 + C_2$ is the combined constant of integration. For definite integrals, the constant of integration cancels out during the evaluation process, so we can ignore it for now.

Step 2: Apply the Fundamental Theorem of Calculus

The fundamental theorem of calculus provides a method to evaluate definite integrals using the antiderivative. It states that if $F(x)$ is an antiderivative of $f(x)$, then:

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

In our case, $f(x) = -1 + x^2$, and we found its antiderivative to be $F(x) = -x + \frac{x^3}{3}$. The limits of integration are $1$ and $a$. Applying the fundamental theorem, we have:

$$\int_1^a [-1 + x^2] dx = F(a) - F(1)$$

First, we evaluate $F(a)$:

$$F(a) = -a + \frac{a^3}{3}$$

Next, we evaluate $F(1)$:

$$F(1) = -1 + \frac{1^3}{3} = -1 + \frac{1}{3} = -\frac{2}{3}$$

Now, we subtract $F(1)$ from $F(a)$:

$$F(a) - F(1) = \left(-a + \frac{a^3}{3}\right) - \left(-\frac{2}{3}\right) = -a + \frac{a^3}{3} + \frac{2}{3}$$

So, the value of the definite integral is:

$$\int_1^a [-1 + x^2] dx = -a + \frac{a^3}{3} + \frac{2}{3}$$

Step 3: Determine the Value of 'a'

To obtain a numerical answer, we need to determine the value of $a$. The problem statement gives us the function $f:[-1,1] \rightarrow [0,1], f(x) = x^2$. This information suggests that the domain of $f(x)$ is $[-1, 1]$ and the range is $[0, 1]$. However, this doesn't directly give us the value of $a$ in the integral. Looking back at the original options provided, we see options with numerical values. This indicates that there may be a missing piece of information in the problem statement, or we need to make a specific choice for $a$ to match the given options.

Let's examine the options provided:

A) $\frac{2}{3}$ B) $\frac{3}{2}$ C) $\frac{7}{6}$ D) $\frac{8}{3}$ E) $\frac{17}{2}$

We need to find a value of $a$ such that:

$$-a + \frac{a^3}{3} + \frac{2}{3} = \text{one of the options}$$

Without additional information, we can try to find a value of $a$ that simplifies the expression. A logical choice might be $a = 1$, which is the upper bound of the domain of $f(x)$. If $a = 1$, the integral becomes:

$$\int_1^1 [-1 + x^2] dx$$

When the limits of integration are the same, the definite integral is zero. However, this doesn't match any of the options. Another approach is to assume that the question intends for us to find the integral given a specific value for $a$ that leads to one of the answers. This might involve working backward from the answer choices or recognizing a particular value of $a$ that simplifies the equation.

Let's consider the possibility that the question is designed such that $a=2$. Substituting $a=2$ into our result, we get:

$$-2 + \frac{2^3}{3} + \frac{2}{3} = -2 + \frac{8}{3} + \frac{2}{3} = -2 + \frac{10}{3} = \frac{-6 + 10}{3} = \frac{4}{3}$$

This value is not among the provided options either. It seems we need to reconsider our approach or the missing information.

Reconsidering the Problem

Given the options and the structure of the problem, it's likely there's a specific context or a missing equation that would allow us to determine the value of $a$. The problem might be part of a larger question, or there might be a condition that we haven't considered. Without this additional information, we can't definitively select one of the provided answers.

However, let's try a different approach. Suppose the problem intended to evaluate the integral:

$$\int_{-1}^1 f(x) dx = \int_{-1}^1 x^2 dx$$

In this case, we would have:

$$\int_{-1}^1 x^2 dx = \left[\frac{x^3}{3}\right]_{-1}^1 = \frac{1^3}{3} - \frac{(-1)^3}{3} = \frac{1}{3} - \left(-\frac{1}{3}\right) = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}$$

This result matches option A. If we were to assume that the intended integral was $\int_{-1}^1 x^2 dx$, then option A would be the correct answer. However, this is a significant deviation from the original problem statement.

Conclusion

Based on our assumption that $h(x) = x^2$, we found the definite integral to be:

$$\int_1^a [-1 + x^2] dx = -a + \frac{a^3}{3} + \frac{2}{3}$$

However, we couldn't determine a specific numerical value without additional information about $a$. If we assume the problem intended to evaluate $\int_{-1}^1 x^2 dx$, then the answer would be $\frac{2}{3}$, which corresponds to option A. But without further clarification, we can't definitively say that this is the intended solution. This detailed analysis highlights the importance of clear problem statements and complete information in mathematics.

Key Takeaways:

• Definite integrals represent the area under a curve between two limits.
• The fundamental theorem of calculus is used to evaluate definite integrals.
• Missing information in a problem statement can lead to ambiguity in the solution.
• Assumptions can be made to proceed with a problem, but they need to be clearly stated.

In summary, while we've explored the process of solving definite integrals, the specific problem requires further clarification to arrive at a definitive answer. We’ve detailed each step and highlighted the importance of having all necessary information to solve mathematical problems accurately.

Repair Input Keyword

What is the value of the definite integral $\int_1^a [-1 + h(x)]dx$, given that $f(x) = x^2$ and assuming $h(x) = f(x)$?

Title

Solving Definite Integrals Example f(x) = x^2