Problem 1 - Calculus Single Variable

Question

Suppose that \(\int_{1}^{3} f(x) d x=-8\) and \(\int_{3}^{7} f(x) d x=12\). Evaluate \(\int_{1}^{7} f(x) d x\)

Step-by-Step Solution

Answer

The value of \(\int_{1}^{7} f(x) dx\) is 4.

Step 1: Understand the problem

We are given two definite integrals: \(\int_{1}^{3} f(x) dx = -8\) and \(\int_{3}^{7} f(x) dx = 12\). The task is to find the value of \(\int_{1}^{7} f(x) dx\) using these given integrals.

Step 2: Apply the property of integrals

Recognize that the integral from 1 to 7 can be split into two separate integrals. Using the property of integrals, \(\int_{1}^{7} f(x) dx = \int_{1}^{3} f(x) dx + \int_{3}^{7} f(x) dx\). This allows us to utilize the given values directly.

Step 3: Substitute known values

Substitute the known values of the integrals given in the problem into the expression from Step 2. We have \(\int_{1}^{3} f(x) dx = -8\) and \(\int_{3}^{7} f(x) dx = 12\). Thus, \(\int_{1}^{7} f(x) dx = -8 + 12\).

Step 4: Calculate the result

Perform the arithmetic operation: \(-8 + 12 = 4\). Thus, the integral \(\int_{1}^{7} f(x) dx = 4\).

Key Concepts

Integral Properties

Understanding integral properties is crucial when tackling calculus problems, especially those involving definite integrals. One key property to know is the **additive property of integrals**, which allows us to combine integrals over adjacent intervals into one overarching integral. For instance, if you have two integrals

• defined over the ranges from 1 to 3 and from 3 to 7, they can be combined.
• This gives us the integral from 1 to 7, as shown by: \[ \int_{1}^{7} f(x) \, dx = \int_{1}^{3} f(x) \, dx + \int_{3}^{7} f(x) \, dx \]

This property is invaluable in simplifying problems because it allows direct integration of known values over contiguous segments.
By understanding this property, you can seamlessly piece together separate integrals to solve more complex calculus problems.

Calculus Problem Solving

Solving calculus problems, particularly those involving integrals, requires a strategic approach. Here's a simplified approach to addressing them:

• First, **understand the problem and the values involved**. Take note of initial conditions, such as the given integral values in the problem: \( \int_{1}^{3} f(x) \, dx = -8 \) and \( \int_{3}^{7} f(x) \, dx = 12 \).
• Next, **employ key calculus properties**. As discussed, use the additive nature of integrals to combine known information: \[\int_{1}^{7} f(x) \, dx = \int_{1}^{3} f(x) \, dx + \int_{3}^{7} f(x) \, dx\]
• Finally, **substitute the known values** and perform the necessary arithmetic operations. By substituting \(-8\) and \(12\) for the integrals, you can simplify:\[-8 + 12 = 4\]

With these steps, complex integrals become easier to manage. This methodical approach aids tremendously in reducing errors and ensures that you're leveraging all available information effectively.

Arithmetic Operations in Calculus

Arithmetic operations are a fundamental component in solving calculus problems, particularly those involving the evaluation of integrals. Once you've expressed the problem using integral properties, the arithmetic operations come into play.

• These operations allow you to simplify and solve expressions that arise from combining integrals.
• Take, for example, the operation involved in combining the integrals’ values: \(-8 + 12\).

Performing this arithmetic leads you to the result of the integral from 1 to 7. Although it might seem straightforward, proper execution of these operations is crucial. Even a simple misstep in arithmetic can lead to incorrect results, masking your understanding of integral properties and calculus problem-solving strategies.
Using the approach highlighted earlier ensures accuracy in both understanding and execution, making calculus problems more accessible.