Problem 89 - Calculus Volume 1

Question

Suppose that \(\int_{0}^{4} f(x) d x=5\) and \(\int_{0}^{2} f(x) d x=-3,\) and \(\int_{0}^{4} g(x) d x=-1\) and \(\int_{0}^{2} g(x) d x=2 .\) In the following exercises, compute the integrals. \(\int_{2}^{4}(f(x)+g(x)) d x\)

Step-by-Step Solution

Answer

The integral \(\int_{2}^{4}(f(x)+g(x)) \, dx\) is 5.

Step 1: Use the Integral Addition Property

We start by acknowledging the property of integration that allows us to separate the integral of a sum: \(\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx\). This means that we can break down the problem into two parts: \(\int_{2}^{4} f(x) \, dx\) and \(\int_{2}^{4} g(x) \, dx\).

Step 2: Apply the Property of Definite Integrals

Using the property of definite integrals, we can express \(\int_{2}^{4} f(x) \, dx\) as \(\int_{0}^{4} f(x) \, dx - \int_{0}^{2} f(x) \, dx\). Similarly, express \(\int_{2}^{4} g(x) \, dx\) as \(\int_{0}^{4} g(x) \, dx - \int_{0}^{2} g(x) \, dx\).

Step 3: Calculate \(\int_{2}^{4} f(x) \, dx\)

Substitute the given values: \(\int_{2}^{4} f(x) \, dx = \int_{0}^{4} f(x) \, dx - \int_{0}^{2} f(x) \, dx = 5 - (-3) = 5 + 3 = 8\).

Step 4: Calculate \(\int_{2}^{4} g(x) \, dx\)

Substitute the given values: \(\int_{2}^{4} g(x) \, dx = \int_{0}^{4} g(x) \, dx - \int_{0}^{2} g(x) \, dx = -1 - 2 = -3\).

Step 5: Sum the Results

Now add the two results from Steps 3 and 4:\(\int_{2}^{4}(f(x) + g(x)) \, dx = \int_{2}^{4} f(x) \, dx + \int_{2}^{4} g(x) \, dx = 8 + (-3) = 5\).

Key Concepts

Integral Addition Property

The Integral Addition Property is a fundamental concept in calculus. It allows you to split the integral of a sum of functions into the sum of individual integrals. This is especially handy when dealing with complex expressions. Let's break it down!

• When you have an integral such as \(\int (f(x) + g(x)) \, dx\), you can rewrite it as \(\int f(x) \, dx + \int g(x) \, dx\).
• This property works for both definite and indefinite integrals, simplifying calculations.
• It's especially useful when you have bounds on the integral like \(\int_{2}^{4}(f(x) + g(x)) \, dx\).

By understanding this property, you can tackle integrals that initially seem daunting by breaking them into more manageable pieces. This approach was the first step in solving the exercise, allowing the integrals of \(f(x)\) and \(g(x)\) to be calculated separately.

Properties of Definite Integrals

Definite integrals have specific properties that can be harnessed to simplify complex integral calculations. Let's explore key aspects that were pivotal in solving the given exercise.

• One notable property is the ability to compute the definite integral over a specific interval by subtracting two integrals. It follows the formula: \(\int_{a}^{b} f(x) \, dx = \int_{c}^{b} f(x) \, dx - \int_{c}^{a} f(x) \, dx\).
• This means you can find \(\int_{2}^{4} f(x) \, dx\) by using known integrals like \(\int_{0}^{4} f(x) \, dx\) and \(\int_{0}^{2} f(x) \, dx\).
• The ability to manipulate the limits of integration using basic arithmetic is crucial for breaking down problems.

In the exercise, this property enabled the calculation of each integral over the desired interval (from 2 to 4). It demonstrates how understanding and applying these properties can make solving definite integrals more straightforward.

Step-by-Step Calculation

Approaching each integral problem methodically through a step-by-step calculation strategy is efficient and ensures accuracy. Here’s how the exercise is solved stepwise:

Step 1: Apply the Integral Addition Property

The problem is broken into two separate integrals by applying the addition property: \(\int_{2}^{4} f(x) \, dx\) and \(\int_{2}^{4} g(x) \, dx\).

Step 2: Use Properties of Definite Integrals

Calculate each integral over \(2\) to \(4\) by subtracting known values: \(\int_{2}^{4} f(x) \, dx\) using \(\int_{0}^{4} f(x) \, dx\) and \(\int_{0}^{2} f(x) \, dx\); similarly for \(g(x)\). The calculations are:

• For \(f(x)\): \(\int_{2}^{4} f(x) \, dx = 5 - (-3) = 8\)
• For \(g(x)\): \(\int_{2}^{4} g(x) \, dx = -1 - 2 = -3\)

Step 3: Sum the Results

Finally, add the integrals from steps 3 and 4 to find \(\int_{2}^{4}(f(x) + g(x)) \, dx = 8 + (-3) = 5\).
This step-by-step approach clarifies how to use integral properties efficiently, ensuring each calculation builds logically on the last, leading neatly to the final result.