Problem 3 - Calculus Single Variable

Question

Suppose that \(\int_{-7}^{3} f(x) d x=-7\) and \(\int_{-7}^{3} g(x) d x=-4\). Evaluate \(\int_{-7}^{3}(4 f(x)-9 g(x)) d x\).

Step-by-Step Solution

Answer

The value of the integral is 8.

Step 1: Understanding the Problem

We need to evaluate the integral \( \int_{-7}^{3}(4 f(x) - 9 g(x)) \, dx \) by using the given values of the integrals of \( f(x) \) and \( g(x) \) over the same interval.

Step 2: Using Linearity of Integration

We can separate the integral \( \int_{-7}^{3}(4 f(x) - 9 g(x)) \, dx \) into two separate integrals: \( 4 \int_{-7}^{3} f(x) \, dx - 9 \int_{-7}^{3} g(x) \, dx \), due to the linearity property of integrals.

Step 3: Substituting Known Values

Given the values \( \int_{-7}^{3} f(x) \, dx = -7 \) and \( \int_{-7}^{3} g(x) \, dx = -4 \), substitute into the expression to get:\[4(-7) - 9(-4).\]

Step 4: Calculating the Expression

Calculate the expression:First, \( 4(-7) = -28 \).Second, \( 9(-4) = -36 \).Now combine: \( -28 - (-36) = -28 + 36 = 8 \).

Key Concepts

Linearity of Integration

Understanding linearity of integration is key in solving integrals involving sums or differences of functions. This principle allows us to simplify complex expressions by breaking them into simpler parts. According to linearity, for two functions, say \( f(x) \) and \( g(x) \), we have:

• Linearity with respect to addition: \( \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \)
• Linearity with respect to multiplication by a constant: \( \int c \cdot f(x) \, dx = c \cdot \int f(x) \, dx \)

These properties simplify the evaluation of integrals by allowing you to take constants out of the integral and evaluate each function's integral separately.
In the original problem, we used these properties to transform the integral: \( \int_{-7}^{3} (4f(x) - 9g(x)) \, dx \) into two separate integrals: \( 4 \int_{-7}^{3} f(x) \, dx - 9 \int_{-7}^{3} g(x) \, dx \).
This makes it easy to apply the known values of each integral, streamlining the solution process.

Integral Properties

Integral properties not only include linearity but other essential rules for solving integrals as well.
Here are some fundamental properties to keep in mind:

• Constant Rule: \( \int c \, dx = c \cdot x + C \), where \( C \) is the constant of integration.
• Addition/Subtraction Rule: This extends the linearity principle to any functions being added or subtracted.
• Bounds of Integration: If you know \( \int_{a}^{b} f(x) \, dx \), reversing the bounds of integration changes the sign: \( \int_{b}^{a} f(x) \, dx = -\int_{a}^{b} f(x) \, dx \).

These properties are the backbone of evaluating definite integrals, which involve calculating the area under the curve from one point to another.
In the given problem scenario, we specifically utilize the linearity and the known integral values to solve quickly and accurately. Integral properties such as symmetry and bounds can also guide the intuition of problem-solving.
Keep these in your toolkit while working with definite integrals to ease up complex calculus problems.

Problem Solving in Calculus

In calculus, integrating functions is a fundamental problem-solving technique.
When confronted with integrals, using a systematic method like the one described can simplify complex problems to straightforward arithmetic.
Here's a quick checklist for tackling similar problems:

• Identify given integral values or properties.
• Apply linearity and separation of integrals, if applicable.
• Use known integral properties such as symmetry or bounds where possible.
• Substitute values and calculate each separate integral part.
• Combine results for the final solution.

In our original exercise, breaking down the problem into steps—a common method in calculus—makes even intimidating problems manageable.
It's about taking advantage of known integral values and properties to substitute and solve.
This approach not only aids in problem-solving but reinforces understanding of core calculus concepts as you practice.