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Chapter 6: Problem 3

Evaluate the definite integrals. \(\int_{-2}^{-2} 3 x^{4} d x\)

Short Answer

Expert verified
The value of the integral is 0 because the limits are the same.

Step by step solution

01

Understanding the Definite Integral

Recognize the definite integral you need to evaluate: \[\int_{-2}^{-2} 3x^4 \, dx\]This integral has identical upper and lower limits, indicating no area under the curve. Let's explore what this means.
02

Applying the Identity of Zero Interval

In calculus, when the limits of integration are the same (i.e., both the upper and lower limits are identical), the definite integral equals zero, regardless of the function being integrated.
03

Evaluate the Integral

Using the principle that the integral over an interval with the same start and end points is zero:\[\int_{-2}^{-2} 3x^4 \, dx = 0\]Thus, the value of this integral is zero.

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Key Concepts

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

Calulus and Its Role in Integration
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It plays a crucial role in understanding change and motion. Integration, a fundamental concept in calculus, helps us find quantities like area under a curve, accumulated change, and total growth.

When tackling problems involving integration, we're usually interested in finding the integral of a function across a certain interval. This interval is defined by two points, the lower limit and the upper limit. The result of taking an integral over this interval gives us the accumulation of the function's values, representing things like total distance traveled or the area under a curve.
Understanding Integration Properties
Integration properties come in handy when solving integrals, as they simplify the process and offer insights into the behavior of integrals.
  • Constant Multiple Rule: If you have an integral like \(\int a \, f(x) \, dx,\), you can factor out the constant \(a\), making it \(a \int f(x) \, dx\). This eases calculations.
  • Sum Rule: The integral of a sum of functions is the sum of their integrals: \(\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx\).
  • Interval Adjustments: Adjusting the limits of integration requires careful attention. Changing these can alter the integral's result significantly.
These properties act as tools for breaking down complex problems into manageable calculations, making integration more approachable.
Zero Interval in Integration
A fascinating aspect of integration involves intervals where the upper and lower limits are identical. This scenario leads to a zero interval. In the case of definite integrals, if both limits are the same, the integral value is zero.

This might initially seem odd because we usually expect a sum. However, since the area under any function from one point to the same point is nonexistent, the result intuitively becomes zero.
  • This applies to any function, no matter its complexity or behavior.
  • For example, \(\int_{-2}^{-2} 3x^4 \, dx \) equals zero, because it spans no interval.
Understanding this principle streamlines evaluating integrals, saving time and effort when faced with identical limits.

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Most popular questions from this chapter

Use the midpoint rule for \(n=1000\) to approximate \(\int_{1}^{3} \frac{1}{x} d x .\) Compare your answer to the correct answer \(\ln 3\) and to the left- and right-hand sums.

Appreciation A painting by one of the masters is purchased by a museum for \(\$ 1,000,000\) and increases in value at a rate given by \(V^{\prime}(t)=100 e^{t},\) where \(t\) is measured in years from the time of purchase. What will the painting be worth in 10 years?

Find the indefinite integral. $$ \int \frac{1}{x \ln \sqrt{x}} d x $$

Explain in complete sentences the difference between an antiderivative of \(f(x)\) and the indefinite integral \(\int f(x) d x\)

The following table gives the earnings in dollars of Swedish households in earnings quartiles. (See De Nardi and colleagues. \({ }^{15}\) ) $$\begin{array}{cccc}\text { 1st } & \text { 2nd } & \text { 3rd } & \text { 4th } \\ 7010 & 28,120 & 44,315 & 76,646\end{array}$$ Determine the coefficient of inequality by taking the average of the right- and left-hand sums.
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