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Chapter 7: Problem 4

Evaluate the definite integral: \(\int_{0}^{2 \pi} \sin (x) \mathrm{d} x\).

Short Answer

Expert verified
The definite integral: \( \int_{0}^{2\pi} \sin(x) \mathrm{d} x = 0 \).

Step by step solution

01

Find the antiderivative

First, we need to find the antiderivative (also known as the integral) of the sin function. The antiderivative of \( \sin(x) \) is \(-\cos(x)\). So, we apply this rule to find the integral of the function: \( \int \sin(x) \mathrm{d} x = -\cos(x) + C\), where C is the constant of integration.
02

Apply the limits of the definite integral

We then replace \(x\) with the limits of the definite integral. First, apply the upper limit \(2\pi\): \(-\cos(2\pi) = -1\), and then subtract the result of applying the lower limit \(-\cos(0) = -1\).
03

Evaluate the definite integral

The next step is to subtract the results obtained from applying the upper and lower limits: \(-1 - (-1) = -1 +1 = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Antiderivatives
Antiderivatives are the reverse of derivatives. When you find an antiderivative, you're essentially undoing the derivative process. This can also be described as finding an indefinite integral. For example, if you know that the derivative of some function is \(-f(x)\), the antiderivative is the function you're looking for. In our exercise, the function is \(\sin(x)\). By finding its antiderivative, we determined it to be \(-\cos(x)\).
  • Antiderivatives help find the area under curves for continuous functions.
  • The general form is \(F(x) + C\), where \(C\) represents any constant.
  • This is essential for further evaluating definite integrals.
Keep in mind that an antiderivative differs from a derivative by a constant, which does not affect integration over a fixed interval.
Evaluating Definite Integrals
Definite integrals are about finding the net area under a curve between specific limits. Unlike indefinite integrals, they give a numerical value, which can represent various real-world quantities like displacement or total change. In the exercise, we evaluated \(\int_{0}^{2\pi} \sin(x) \, \mathrm{d}x\).

Here's the process of evaluating definite integrals:
  • Find the antiderivative of the function.
  • Apply the upper limit to the antiderivative, then the lower limit.
  • Subtract the value of the lower limit from the upper limit.
The subtraction in our case resulted in \(-1 - (-1) = 0\), showing no net area, meaning the areas above and below the x-axis cancel each other. This emphasizes the harmony in periodic functions over their full periods when evaluated.
Exploring Trigonometric Integrals
Trigonometric integrals are integrals involving trigonometric functions like sine, cosine, and tangent. They often require special strategies and identities. In this case, we dealt with \(\sin(x)\), which cycles between -1 and 1.
  • The antiderivative of \(\sin(x)\) is \(-\cos(x)\).
  • Understanding the properties of trig functions helps in simplifying integrals effectively.
  • Useful properties like periodicity can simplify some integrations significantly.
Trigonometric integrals are common in physics and engineering because these functions model waves and oscillations. Recognizing patterns, like symmetry and periodicity, can greatly streamline the calculation process.

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Most popular questions from this chapter

Find the indefinite integral without using a table: (a) \(\int x \ln (x) \mathrm{d} x\). (b) \(\int x \sin ^{2}(x) \mathrm{d} x\).

Determine whether the following improper integrals converge. Evaluate the convergent integrals. (a) \(\int_{1}^{\infty}\left(\frac{1}{x^{2}}\right) d x\). (b) \(\int_{1}^{\pi / 2} \tan (x) \mathrm{d} x\).

Find the indefinite integral without using a table: \(\int \frac{1}{x(x-a)} d x\).

Find the following area by computing the values of a definite integral: The area bounded by the straight line \(y=2 x+3\), the \(x\) axis, the line \(x=1\), and the line \(x=4\).

Determine whether the following improper integrals converge. Evaluate the convergent integrals (a) \(\int_{0}^{1} \frac{1}{x \ln (x)} \mathrm{d} x\). (b) \(\int_{1}^{\infty}\left(\frac{1}{x}\right) \mathrm{d} x\).
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