Problem 81 - Calculus for Biology and Medicine

Question

Find the value of \(a \in[0,2 \pi]\) that maximizes \(\int_{0}^{a} \cos x d x\).

Step-by-Step Solution

Answer

The value of \( a \) that maximizes the integral is \( \frac{\pi}{2} \).

Step 1: Understand the Integral

The integral \( \int_{0}^{a} \cos x \, dx \) represents the area under the curve \( \cos x \) from \( x = 0 \) to \( x = a \). Our task is to find the value of \( a \) within the interval \([0, 2\pi]\) that maximizes this integral.

Step 2: Find the Indefinite Integral

The indefinite integral of \( \cos x \) is \( \int \cos x \, dx = \sin x + C \).

Step 3: Evaluate the Definite Integral

Now we can use the definite integral to express \( F(a) = \int_{0}^{a} \cos x \, dx = \sin a - \sin 0 = \sin a \) since the integral evaluates to \( \sin a \) with \( \sin 0 = 0 \).

Step 4: Analyze the Function \( \sin a \)

The function \( \sin a \) reaches its maximum value of 1 at \( a = \frac{\pi}{2} \), as it increases from 0 to \( \frac{\pi}{2} \) and decreases afterward on the interval \([0, 2\pi]\). Thus, the maximum value of the integral occurs at this point.

Key Concepts

Definite Integral

A definite integral provides a precise mathematical way to calculate the area under a curve between two specific points.
When you see \[ \int_{0}^{a} \cos x \, dx \]it means we are calculating the area from \(x=0\) to \(x=a\) under the curve of the function \( \cos x \). This integral is considered "definite" because it has clear limits of integration, namely 0 and \(a\).

• The lower limit (0) marks where you start measuring the area.
• The upper limit (\(a\)) marks where you stop measuring the area.

In our exercise's case, finding the value of \(a\) that maximizes the integral means determining which segment under the sine wave yields the largest area. This process highlights the relationship between the geometry of the trigonometric function and the calculus concepts used to measure it.

Indefinite Integral

An indefinite integral, sometimes called an antiderivative, is the reverse of taking the derivative. It gives us a family of functions rather than a specific numerical value. For the function \( \cos x \), the indefinite integral is:\[\int \cos x \, dx = \sin x + C\]

• \( \sin x \) is the antiderivative or indefinite integral of \( \cos x \).
• \( C \) represents the constant of integration.

The indefinite integral is helpful in solving definite integrals because it allows us to establish a general formula that we can apply to specific limits. In this exercise, we evaluated the definite integral between the fixed limits 0 and \(a\) by first determining the indefinite integral. This led us to the function \( \sin a \), from which we can easily find the maximum value by analyzing its behavior over the defined interval.

Trigonometric Functions

Trigonometric functions, such as \( \cos x \) and \( \sin x \), are mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are periodic, meaning they repeat their values in regular intervals, which is highly relevant in calculus.

• \( \cos x \) starts at 1 when \(x = 0\), decreases to -1 by \(x = \pi\), and returns back to 1 at \(x = 2\pi\).
• \( \sin x \) starts from 0, reaches its maximum of 1 at \(x = \pi/2\), and oscillates between -1 and 1.

In our task, understanding the behavior of the \( \sin a \) function is crucial as it decides the maximum area. We discovered that \( \sin a \) reaches its peak within the interval \([0, 2\pi]\) at \(a=\frac{\pi}{2}\). Trigonometric functions' periodic and oscillating nature allows us to calculate these critical points, providing insight into real-world cyclic phenomena.