Problem 10 - Calculus Single Variable

Question

Suppose that \(\int_{2}^{-9} f(x) d x=5\). Evaluate \(\int_{-9}^{2}(3 f(x)-5 x) d x\).

Step-by-Step Solution

Answer

The evaluated integral is 177.5.

Step 1: Understanding the Problem

We are given the integral \( \int_{2}^{-9} f(x) \, dx = 5 \) and asked to evaluate the integral \( \int_{-9}^{2} (3f(x) - 5x) \, dx \). Recognize that the limits of integration are reversed in the given integral, which affects our calculations.

Step 2: Reverse the Integration Order

To reverse the limits of the given integral, we apply the property \( \int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx \). This means \( \int_{-9}^{2} f(x) \, dx = -5 \).

Step 3: Apply Linearity of Integration

Use the linearity property of integrals which states \( \int_{a}^{b} [cf(x) + dg(x)] \, dx = c\int_{a}^{b} f(x) \, dx + d\int_{a}^{b} g(x) \, dx \). Thus, rewrite the problem: \( \int_{-9}^{2} [3f(x) - 5x] \, dx = 3\int_{-9}^{2} f(x) \, dx - 5\int_{-9}^{2} x \, dx \).

Step 4: Evaluate the Integral of \( 3f(x) \)

Substitute the calculated value of \( \int_{-9}^{2} f(x) \, dx = -5 \) into the expression: \( 3 \times (-5) = -15 \).

Step 5: Evaluate the Integral of \( -5x \)

Compute \( -5 \int_{-9}^{2} x \, dx \). First, find \( \int_{-9}^{2} x \, dx \) which is \( \left[ \frac{x^2}{2} \right]_{-9}^{2} \). Substitute: \( \frac{2^2}{2} - \frac{(-9)^2}{2} = 2 - 40.5 = -38.5 \). So, \(-5 \times -38.5 = 192.5 \).

Step 6: Combine the Results

Add the results from Steps 4 and 5: \(-15 + 192.5 = 177.5 \). This is the evaluated integral \( \int_{-9}^{2} (3f(x) - 5x) \, dx \).

Key Concepts

Reversing Limits of Integration

When tackling an integral with limits that appear reversed, it can initially be confusing. However, there is a straightforward solution: switch the limits to reorient the integral to a more familiar form. This process hinges on a special property of definite integrals. By mathematical definition, reversing the limits of an integral changes the sign. Specifically, \( \int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx \). This rule can be particularly useful for simplifying calculations.

For example, if you are given \( \int_{2}^{-9} f(x) \, dx = 5 \), you can determine that \( \int_{-9}^{2} f(x) \, dx = -5 \). Always remember this switch not only alters the calculation but emphasizes the flexibility of integrals. When flipping limits, think of it as flipping signs. This valuable technique can simplify the execution of computations, ensuring clarity and accuracy.

Linearity of Integration

The linearity of integration is a fundamental principle in calculus. It lies at the heart of many integration techniques. This property enables us to manage complex integrals by distributing them across sums and differences, and even factoring out constants. It can be expressed as \( \int_{a}^{b} [cf(x) + dg(x)] \, dx = c\int_{a}^{b} f(x) \, dx + d\int_{a}^{b} g(x) \, dx \).

Here are some ways linearity provides a powerful toolset:

• Constant Multipliers: You can pull constant coefficients out of an integral.
• Break Apart Sums/Differences: Integrals of sums can be split into the sum of integrals, making them more straightforward to evaluate.

In practice, suppose you need to evaluate \( \int_{-9}^{2} (3f(x) - 5x) \, dx \). Applying the linearity principle simplifies this to \( 3\int_{-9}^{2} f(x) \, dx - 5\int_{-9}^{2} x \, dx \), breaking it down into simpler tasks. Linearity transforms daunting problems into manageable pieces.

Integral Evaluation and Solution Combination

Evaluating integrals, especially using properties like reversing limits and linearity, is an art of combining solutions. After simplifying an expression using these properties, we often arrive at straightforward integrals that need computation.

Consider our example where \( \int_{-9}^{2} f(x) \, dx = -5 \) and you proceed with evaluating \( 3\int_{-9}^{2} f(x) \, dx \). Here you simply multiply by the constant: \( 3 \times (-5) = -15 \).

Next, evaluate \( -5\int_{-9}^{2} x \, dx \). Integrate \( x \) to get \( \left[ \frac{x^2}{2} \right]_{-9}^{2} \), which yields the antiderivative difference: \(2 - 40.5 = -38.5 \). Consequently, \(-5 \times -38.5 = 192.5 \).

Finally, combining these results is much like solving a puzzle. Add \(-15 + 192.5 \) to find the result \( 177.5 \). This systematic approach, where small evaluated pieces add up to a complete answer, ensures accuracy and builds intuition for handling integrals.