Problem 40 - Biocalculus Calculus for the Life Sciences

Question

Find $$\int_{0}^{5} f(x) d x$$ if $$f(x)=\left\\{\begin{array}{ll}{3} & {\text { for } x<3} \\ {x} & {\text { for } x \geqslant 3}\end{array}\right.$$

Step-by-Step Solution

Answer

The integral is 17.

Step 1: Interpret the Function

The function \( f(x) \) is a piecewise function. It is given by \( f(x) = 3 \) for \( x < 3 \) and \( f(x) = x \) for \( x \geqslant 3 \). This means that the function changes its form at \( x = 3 \).

Step 2: Set Up the Integral

Since the function changes at \( x = 3 \), split the integral into two parts: \( \int_{0}^{3} f(x) \, dx \) and \( \int_{3}^{5} f(x) \, dx \). These need to be calculated separately.

Step 3: Evaluate the First Integral

For \( x < 3 \), \( f(x) = 3 \). Thus, calculate \( \int_{0}^{3} 3 \, dx \).\[\int_{0}^{3} 3 \, dx = 3x \bigg|_0^3 = 3 \times 3 - 3 \times 0 = 9\]

Step 4: Evaluate the Second Integral

For \( x \geq 3 \), we have \( f(x) = x \). Thus, calculate \( \int_{3}^{5} x \, dx \).\[\int_{3}^{5} x \, dx = \frac{x^2}{2} \bigg|_3^5 = \frac{5^2}{2} - \frac{3^2}{2} = \frac{25}{2} - \frac{9}{2} = \frac{16}{2} = 8\]

Step 5: Add the Results

Add the results of the two integrals to find the total area under the curve from \( x = 0 \) to \( x = 5 \).\[\int_{0}^{5} f(x) \, dx = 9 + 8 = 17\]

Key Concepts

Definite Integral

Understanding the concept of a definite integral is vital when dealing with calculus problems. The definite integral, represented as \( \int_{a}^{b} f(x) \, dx \), calculates the accumulated total or the "net signed area" between a function's graph and the x-axis, from one point ("a") to another ("b"). Unlike an indefinite integral that results in a function plus a constant, a definite integral evaluates to a specific number.

Some important points to understand about definite integrals are:

• The limits of integration "a" and "b" are crucial as they set the interval over which the function is being integrated.
• "f(x)" is the integrand, which is the function we're accumulating the area for.
• If the curve lies above the x-axis, the area is positive; if it lies below, the area is negative.
• The Fundamental Theorem of Calculus plays a crucial role, connecting the process of finding antiderivatives to evaluating definite integrals.

Thus, a definite integral gives a precise measure of how much the function accumulates within the bounds defined by "a" and "b".

Integration of Piecewise Functions

Piecewise functions are those which have different expressions for various intervals of the independent variable. Integration of such functions requires splitting the overall integral into parts that correspond with these intervals. This is because the rule for the function changes depending on the value of the independent variable, and each piece needs to be integrated with respect to its specific rule.

Let’s see how this is done step by step:

• First, identify where the function changes its expression or rule. In our exercise, this happens when \( x = 3 \).
• Next, set up separate integrals for each segment determined by these change points. For example, \( \int_{0}^{3} f(x)\, dx \) for the first rule and \( \int_{3}^{5} f(x)\, dx \) for the second.
• Finally, calculate each smaller integral using standard integration techniques and add up these results to find the overall integral over the desired interval.

This methodology of breaking up an integral based on the behavior of a piecewise function ensures that the entire area under the curve is accurately calculated.

Area Under the Curve

The area under the curve is essentially what we're looking for when evaluating a definite integral in calculus. This concept is particularly helpful in real-world applications where it represents quantities like distance traveled, total accumulation, or even simple geometric areas.

When dealing with piecewise functions, finding the area under the curve involves integrating each segment separately and adding the results together. Here’s where it’s important:

• It represents the total "net area" accumulated between the function's curve and the x-axis within given limits.
• It applies the integral rules for each function segment to ensure that the results are accurate for each different part of the curve.
• The area might involve combining positive and negative results, depending on whether parts of the curve lie above or below the x-axis.

By understanding the area under the curve, especially with piecewise functions, we gain important insights into the behavior of physical systems or data being analyzed over a specified range.