Problem 106
Question
Evaluate the definite integrals. $$ \int_{-\pi / 3}^{\pi / 3} 2 \cos \left(\frac{x}{2}\right) d x $$
Step-by-Step Solution
Verified Answer
The value of the definite integral is 4.
1Step 1: Identify the Integral
We need to evaluate the definite integral \( \int_{-\pi/3}^{\pi/3} 2 \cos \left(\frac{x}{2}\right) \ dx \). The integral represents the area under the curve of \(2 \cos \left(\frac{x}{2}\right)\) from \(-\pi/3\) to \(\pi/3\).
2Step 2: Find the Antiderivative
To evaluate the integral, we first find the antiderivative of the function. The function is \(2 \cos \left(\frac{x}{2}\right)\). The antiderivative of \(\cos\left(\frac{x}{2}\right)\) with respect to \(x\) is \(2 \sin\left(\frac{x}{2}\right)\) because of the chain rule, multiplying by the reciprocal of \(\frac{1}{2}\). Hence, the antiderivative is \(4 \sin\left(\frac{x}{2}\right)\).
3Step 3: Evaluate the Antiderivative at Boundaries
Substitute the upper limit \(x = \pi/3\) into the antiderivative: \[ 4 \sin\left(\frac{\pi}{6}\right) = 4 \times \frac{1}{2} = 2 \]. Substitute the lower limit \(x = -\pi/3\) into the antiderivative: \[ 4 \sin\left(-\frac{\pi}{6}\right) = 4 \times \left(-\frac{1}{2}\right) = -2 \].
4Step 4: Calculate the Definite Integral
Compute the difference between the results at the upper and lower limits: \[ 2 - (-2) = 2 + 2 = 4 \]. This result represents the total area under the function from \(-\pi/3\) to \(\pi/3\).
Key Concepts
AntiderivativeCosine FunctionArea Under the Curve
Antiderivative
An antiderivative is a function whose derivative gives the original function. It is closely related to finding the area under a curve, known as integration. When you take the antiderivative of a function, you are essentially reversing the process of differentiation.
In our exercise, we are dealing with the function \( 2 \cos \left(\frac{x}{2}\right) \). To find its antiderivative, we rely on the knowledge that the antiderivative of \( \cos\frac{x}{2} \) is \( 2 \sin \frac{x}{2} \), thanks to the chain rule. Why multiply by 2? Because the derivative of \( \sin \left(\frac{x}{2}\right) \) includes \( \frac{1}{2} \), and we need to compensate for this by multiplying the antiderivative by its reciprocal.
In our exercise, we are dealing with the function \( 2 \cos \left(\frac{x}{2}\right) \). To find its antiderivative, we rely on the knowledge that the antiderivative of \( \cos\frac{x}{2} \) is \( 2 \sin \frac{x}{2} \), thanks to the chain rule. Why multiply by 2? Because the derivative of \( \sin \left(\frac{x}{2}\right) \) includes \( \frac{1}{2} \), and we need to compensate for this by multiplying the antiderivative by its reciprocal.
- Antiderivatives help in evaluating definite integrals.
- They are essential in various applications, including physics and engineering.
- Remember to check your work by differentiating back to the original function.
Cosine Function
The cosine function is a fundamental trigonometric function that describes how one side of a right triangle relates to another. For any angle \(x\), the cosine function, written as \(\cos(x)\), varies between -1 and 1, meaning it oscillates in a wave-like pattern.
In our specific integral \( 2 \cos \left( \frac{x}{2} \right) \), the cosine function is scaled and transformed. Scaling involves multiplying by a factor, like 2, which affects the amplitude of the wave, making it taller or shorter. The "\( \frac{x}{2} \)" part compresses the wave, affecting its frequency or how fast it oscillates. These changes can affect the integral's value because they shape the area under the curve.
In our specific integral \( 2 \cos \left( \frac{x}{2} \right) \), the cosine function is scaled and transformed. Scaling involves multiplying by a factor, like 2, which affects the amplitude of the wave, making it taller or shorter. The "\( \frac{x}{2} \)" part compresses the wave, affecting its frequency or how fast it oscillates. These changes can affect the integral's value because they shape the area under the curve.
- Cosine functions are periodic, with a standard period of \( 2\pi \).
- They are crucial in modeling wave behavior, such as light or sound waves.
- In calculus, they are essential in solving integrals and derivatives involving trigonometric functions.
Area Under the Curve
Finding the area under a curve is a key application of integration. This area represents the total accumulated value, such as distance traveled over time, or in our context, the value of the integral.
When dealing with definite integrals like \( \int_{-\pi / 3}^{\pi / 3} 2 \cos \left(\frac{x}{2}\right) dx \), you're essentially finding the area from the curve to the x-axis between two points. The key is using the antiderivative method for evaluation. It requires substituting the limits of integration into the antiderivative, then subtracting the lower limit result from the upper limit result. This process is calculated as follows:- Evaluate the antiderivative at the upper limit.- Evaluate it at the lower limit.- Find the difference.
When dealing with definite integrals like \( \int_{-\pi / 3}^{\pi / 3} 2 \cos \left(\frac{x}{2}\right) dx \), you're essentially finding the area from the curve to the x-axis between two points. The key is using the antiderivative method for evaluation. It requires substituting the limits of integration into the antiderivative, then subtracting the lower limit result from the upper limit result. This process is calculated as follows:- Evaluate the antiderivative at the upper limit.- Evaluate it at the lower limit.- Find the difference.
- This approach ensures you capture the "net" area, accounting for part of the curve below the x-axis.
- It is widely used in physics for calculating work or energy.
- Helps in determining probabilities when dealing with probability density functions in statistics.
Other exercises in this chapter
Problem 104
Evaluate the definite integrals. $$ \int_{-1}^{2}(2+3 t)^{2} d t $$
View solution Problem 105
Evaluate the definite integrals. $$ \int_{0}^{\pi / 4} \sin (2 x) d x $$
View solution Problem 107
Evaluate the definite integrals. $$ \int_{0}^{\pi / 8} \sec ^{2}(2 x) d x $$
View solution Problem 108
Evaluate the definite integrals. $$ \int_{-\pi / 4}^{\pi / 4} \tan x d x $$
View solution