Problem 17 - Applied Calculus

Question

If we know $$\int_{2}^{5} f(x) d x=4,$$ what is the value of $$3\left(\int_{2}^{5} f(x) d x\right)+1 ?$$

Step-by-Step Solution

Answer

The value is 13.

Step 1: Identify the Given Integral

We are given that \( \int_{2}^{5} f(x) \, dx = 4 \). This is the integral of the function \( f(x) \) from 2 to 5.

Step 2: Understand the Expression to Evaluate

We need to evaluate the expression \( 3\left(\int_{2}^{5} f(x) \, dx\right) + 1 \). This means we need to first multiply the given integral by 3 and then add 1.

Step 3: Compute the Multiplication

Multiply the given integral by 3. Since \( \int_{2}^{5} f(x) \, dx = 4 \), we have:\[3 \times 4 = 12\]

Step 4: Add 1 to the Result

After calculating \( 3 \times 4 = 12 \), we add 1 to this result:\[12 + 1 = 13\]

Step 5: Conclusion

The expression \( 3\left(\int_{2}^{5} f(x) \, dx\right) + 1 \) evaluates to 13.

Key Concepts

Integration

Integration is a fundamental concept in calculus, often described as the process of finding the area under a curve. In mathematical terms, an integral represents the accumulation of quantities, such as areas and volumes. The

• "definite integral" is a type of integration that calculates the net area between the curve of a function and the x-axis within a certain interval.
• It is denoted as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration and \( f(x) \) is the function being integrated.

In our exercise, the integral \( \int_{2}^{5} f(x) \, dx = 4 \) informs us that the net area between the curve \( f(x) \) and the x-axis from x = 2 to x = 5 equals 4.
This value is crucial for evaluating further expressions involving this integral.

Calculus Problem Solving

Solving calculus problems, such as those involving definite integrals, requires understanding both the question and the mathematical principles at play. It’s important to break down each step into manageable parts. In the given exercise:

• We first recognize what is known: the integral from 2 to 5 equals 4.
• Next, we understand the task: evaluate the expression \( 3\left(\int_{2}^{5} f(x) \, dx\right) + 1 \).
• We then perform each operation in sequence to reach a final solution.

By systematically applying known values and operations, we can solve calculus problems more efficiently and accurately.

Evaluating Expressions

Once we have the value of a definite integral, we are often required to perform additional calculations using it. In evaluating expressions like the one in our problem, it’s essential to:

• Substitute the known value into the expression accurately. Here, we substitute \( \int_{2}^{5} f(x) \, dx = 4 \) into \( 3\left(\int_{2}^{5} f(x) \, dx\right) + 1 \).
• Perform arithmetic operations step by step. This typically involves multiplication followed by addition or subtraction as per the expression's requirements.
• Arrive at the final value, confirming each step is executed correctly.

For our expression, multiplying the integral's value by 3 resulted in 12, and adding 1 gives us the final evaluated value of 13.
This systematic approach ensures clarity and precision in evaluating mathematical expressions.