Problem 65 - Calculus Single Variable

Question

A sum of integrals of the form \(\int_{a}^{b} f(x) d x\) is given. Express the sum as a single integral of form \(\int_{c}^{d} g(y) d y\). $$ \int_{0}^{2} \sqrt{x} d x=\int_{2}^{4} \sqrt{4-x} d x $$

Step-by-Step Solution

Answer

The sum of integrals can be expressed as \(\int_{0}^{4} g(x) \, dx\) where \(g(x)\) is a piecewise function.

Step 1: Evaluate Each Integral Separately

The given exercise includes two integrals:- The first integral is \( \int_{0}^{2} \sqrt{x} \, dx \).- The second integral is \( \int_{2}^{4} \sqrt{4-x} \, dx \).For now, identify each individual function; here, they are \( \sqrt{x} \) and \( \sqrt{4-x} \). We will ultimately form a new function or integral based on these.

Step 2: Consider the Integrals as Parts of a Piecewise Function

Notice that the intervals are adjacent and cover a continuous range from \(x = 0\) to \(x = 4\). This means we could express these integrals as a single integral of a piecewise function:- \( f(x) = \sqrt{x} \) over \([0, 2] \)- \( f(x) = \sqrt{4-x} \) over \([2, 4] \)Thus, the sum of the integrals can be viewed as a full integral over the interval \([0, 4] \) using this piecewise-defined function \( f(x) \).

Step 3: Write the Single Integral Expression

Using the piecewise-defined function from the previous step, express the original sum of integrals as a single integral:\[\int_{0}^{2} \sqrt{x} \, dx + \int_{2}^{4} \sqrt{4-x} \, dx = \int_{0}^{4} g(x) \, dx\]where \( g(x) = \begin{cases} \sqrt{x}, & \text{if } 0 \le x < 2 \ \sqrt{4-x}, & \text{if } 2 \le x \le 4 \end{cases} \).

Key Concepts

Piecewise Function

A piecewise function is a type of function defined by different expressions over distinct parts of its domain. It allows you to describe multiple behaviors in a single function statement. In the given exercise, we have two integrals that comprise different square root functions over separate intervals. Below are details of how this works:

• From \(x = 0\) to \(x = 2\), the function is \(\sqrt{x}\). This captures the behavior of the function in the first part of the domain.
• From \(x = 2\) to \(x = 4\), the function becomes \(\sqrt{4-x}\). This portion describes the function for the remainder of the domain.

This way, a piecewise function provides a complete picture of a scenario where a simple function isn't sufficient to describe all the behaviors.

Piecewise functions are particularly useful in mathematical modeling, where different rules apply to different conditions or phases.

Definite Integral

The definite integral is a way of calculating the area under a curve defined by a function within a specific interval. It is denoted by the integral symbol \(\int\) followed by bounds that represent the interval over which the function is evaluated.

In this context, the exercise involves calculating the area under two different functions over two intervals, \([0, 2]\) and \([2, 4]\). Each integral separately calculates the area under its respective curve:

• The first integral \(\int_{0}^{2} \sqrt{x} \, dx\) finds the area from \(x = 0\) to \(x = 2\).
• The second integral \(\int_{2}^{4} \sqrt{4-x} \, dx\) finds the area from \(x = 2\) to \(x = 4\).

When separated, these calculations may seem independent, yet the combined perspective unifies them into a single contiguous area from \(x = 0\) to \(x = 4\). This integration over an entire interval is critical in many applications like physics, engineering, and probability.

Integration Techniques

Understanding integration techniques is pivotal in solving problems involving definite integrals, such as the one in the exercise. Different methods might be used based on the function you are trying to integrate:

• Basic Integration: This involves directly integrating basic functions like \(\sqrt{x}\). Here, the power rule is typically applied.
• Substitution: While the functions in this exercise don't necessarily require complicated substitutions, the technique is often utilized if the integrand includes composite functions.
• Piecewise Integration: Specifically in the context of piecewise functions, as in our example, one might handle each segment individually. Subsolutions are computed, then summed for the complete solution.

In this particular exercise, recognizing the piecewise nature of the functions made it simpler to convert the integrals into a single expression. Problems like these instinctively develop your capacity to choose and apply integration techniques efficiently.