Problem 106 - Calculus for Biology and Medicine

Question

Evaluate the definite integrals. $$ \int_{-\pi / 3}^{\pi / 3} 2 \cos \left(\frac{x}{2}\right) d x $$

Step-by-Step Solution

Answer

The value of the definite integral is 4.

Step 1: Identify the Integral to Solve

We need to evaluate the definite integral: \[\int_{-\pi / 3}^{\pi / 3} 2 \cos \left(\frac{x}{2}\right) dx\] This means we aim to find the area under the curve from \(-\pi/3\) to \(\pi/3\).

Step 2: Find the Antiderivative

To solve the integral, we first find the antiderivative of the function. Let's integrate \(2 \cos \left(\frac{x}{2}\right)\):\[\int 2 \cos \left(\frac{x}{2}\right) dx\]By using substitution, we set \(u = \frac{x}{2}\) which means \(du = \frac{1}{2} dx\) or \(dx = 2 du\). Therefore, the integral becomes:\[\int 2 \cos(u) \cdot 2 du = 4 \int \cos(u) du\]The antiderivative of \(\cos(u)\) is \(\sin(u)\), so:\[4 \sin(u) + C = 4 \sin\left(\frac{x}{2}\right) + C\]is the antiderivative.

Step 3: Evaluate the Definite Integral

Now we will evaluate the definite integral by substituting the bounds into our antiderivative:\[4 \sin\left(\frac{x}{2}\right)\Bigg|_{-\pi/3}^{\pi/3}\]Calculate:\[4 \sin\left(\frac{\pi}{6}\right) - 4 \sin\left(-\frac{\pi}{6}\right)\]We know \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\), so it follows that \(\sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2}\).Thus:\[4\left(\frac{1}{2}\right) - 4\left(-\frac{1}{2}\right) = 4\left(\frac{1}{2} + \frac{1}{2}\right)\]Which simplifies to:\[4 \times 1 = 4\]

Step 4: Conclusion

The evaluated definite integral is \(4\). Therefore, the area under the curve \(2\cos\left(\frac{x}{2}\right)\) from \(-\pi/3\) to \(\pi/3\) is \(4\).

Key Concepts

Cosine Function

The cosine function, denoted as \(\cos(x)\), is one of the fundamental trigonometric functions. It describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Cosine is a periodic function with a period of \(2\pi\), meaning its values repeat every \(2\pi\) units.

• Properties:
  • Range: The cosine function's output is always between \(-1\) and \(1\), inclusive.
  • Even Function: Cosine is an even function, which means \(\cos(-x) = \cos(x)\). This property can often simplify integration when dealing with symmetrical bounds, such as from \(-a\) to \(a\), as seen in the given exercise.

The graph of the cosine function is a smooth wave that starts at its maximum (1), dips to -1, and returns to maximum over each period. Knowing the behaviour of cosine helps in evaluating integrals, especially when combined with transformations like translations and dilations.

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function. In calculus, finding the antiderivative is the reverse process of differentiation.

• Basic Concept: If \(F(x)\) is an antiderivative of \(f(x)\), then \(F'(x) = f(x)\). For the cosine function, \(\cos(x)\), its antiderivative is \(\sin(x)\) plus a constant, commonly represented as \(\sin(x) + C\), where \(C\) is an integration constant.
• Importance: Understanding antiderivatives is crucial for solving integrals, especially definite integrals as it allows us to find the enclosed area under a curve
  • This is vital in physics and engineering for calculating quantities like displacement or work.

In the exercise, we calculated the antiderivative of \(2\cos\left(\frac{x}{2}\right)\) by first transforming the variable, which simplified the function inside the integral.

Integration by Substitution

Integration by substitution is a powerful technique used to simplify complex integrals. It's sometimes compared to the reverse of the chain rule in differentiation.

• Substitution Method: This technique involves changing the variable of integration.
  • In the given exercise, we defined a substitution \(u = \frac{x}{2}\). This substitution transforms the integral into a simpler one in terms of \(u\).
  • The derivative \(du = \frac{1}{2}dx\) helps in expressing \(dx\) in terms of \(du\), thereby changing the integral entirely in terms of \(u\): \[dx = 2du\]
• Simplification: The substitution reduces the integral
  • Thus, the original complex integral becomes \(4\int\cos(u)\,du\), which is straightforward to integrate.

Integration by substitution effectively transforms complicated integrals into easier ones, often resulting in straightforward antiderivatives. This technique is a fundamental part of calculus, enabling us to evaluate a wide variety of integrals efficiently.