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Chapter 5: Problem 48

Evaluate the expression, if possible, or say what additional information is needed, given that \(\int_{-4}^{4} g(x) d x=12\). $$\int_{-4}^{4} g(-x) d x$$

Short Answer

Expert verified
The integral \(\int_{-4}^{4} g(-x) \, dx\) is 12.

Step by step solution

01

Recognize Even Function Property

A function is considered even if for all \(x\), \(g(x) = g(-x)\). This means that the function is symmetric with respect to the y-axis. Our goal is to check if the given function \(g(-x)\) has this property.
02

Analyze the Provided Integral

We are given \( \int_{-4}^{4} g(x) \, dx = 12 \). This means that the area under the curve \(g(x)\) from \(-4\) to \(4\) is 12.
03

Relationship between \(g(-x)\) and \(g(x)\)

Consider that for any function \(g(x)\), \(\int_{-a}^{a} g(-x) \, dx\) will have the same value as \(\int_{-a}^{a} g(x) \, dx\), provided that the function \(g(x)\) is even, because \(g(-x) = g(x)\).
04

Evaluate the Integral

Since we have \(\int_{-4}^{4} g(x) \, dx = 12\) and the integral from \(-a\) to \(a\) for an even function has the same value regardless of flipping the sign of \(x\), we find \(\int_{-4}^{4} g(-x) \, dx = 12\).

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Key Concepts

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

Even Functions
Even functions are a special type of function where for every value of \( x \), the function yields the same result for \( -x \). This can be expressed mathematically as \( g(x) = g(-x) \). These functions exhibit symmetry about the y-axis.

Here are some properties of even functions:
  • If you reflect the graph of an even function across the y-axis, it looks unchanged. This is visual symmetry.
  • Typical examples include \( x^2, \, x^4, \, \text{and} \, \cos(x) \).
  • The concept of even functions is crucial when evaluating certain kinds of integrals, as symmetry can simplify the calculations.
Identifying whether a function is even helps in solving integral problems as demonstrated in our original exercise. If the integrand is even over a symmetric interval, calculations become easier due to the function's inherent symmetry.
Definite Integrals
A definite integral is the calculation of the area under a curve between two limits, let's call them \( a \) and \( b \). It is represented as \( \int_{a}^{b} g(x) \, dx \). Definite integrals provide a numerical value that represents this area.

Key aspects of definite integrals:
  • The integral has upper and lower limits, \( a \) and \( b \), which define the start and end points on the x-axis.
  • The result of a definite integral can be a positive or negative number, or even zero, indicating the net area between the curve and the x-axis.
  • Adjustments can be made by calculating from \( -a \) to \( a \) in terms of symmetry, which particularly simplifies when dealing with even functions.
  • In the context of our exercise, realizing \( \int_{-4}^{4} g(x) \, dx = 12 \) provides vital information when determining the symmetric counterpart integral \( \int_{-4}^{4} g(-x) \, dx \).
Understanding definite integrals is essential for calculating areas and solving problems in physics, engineering, and beyond.
Symmetry in Integrals
Symmetry plays an important role in simplifying the computation of integrals. When a function is symmetric, it means that parts of its domain mirror each other in some way, leading to potential simplifications.

For even functions over symmetric limits, such as from \( -a \) to \( a \), this simplification is notably useful. Consider the integral \( \int_{-a}^{a} g(x) \, dx \):
  • If \( g(x) \) is even, then the area under \( g(x) \) from \( -a \) to \( a \) is simply twice the area from \( 0 \) to \( a \). This is because \( g(x) = g(-x) \).
  • For the integral \( \int_{-4}^{4} g(x) \, dx \), symmetry informs us that examining \( \int_{-4}^{4} g(-x) \, dx \) under the same condition does not affect the outcome, hence it also equals 12.
Analyzing such symmetry allows mathematicians to reduce computing complexity, achieving results faster and often using elementary reasoning alone.

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Most popular questions from this chapter

(a) On a sketch of \(y=\ln x,\) represent the left Riemann sum with \(n=2\) approximating \(\int_{1}^{2} \ln x d x .\) Write out the terms in the sum, but do not evaluate it. (b) On another sketch, represent the right Riemann sum with \(n=2\) approximating \(\int_{1}^{2} \ln x d x .\) Write out the terms in the sum, but do not evaluate it. (c) Which sum is an overestimate? Which sum is an underestimate?

(a) Using a graph, decide if the area under \(y=e^{-x^{2} / 2}\) between 0 and 1 is more or less than 1 (b) Find the area.

Evaluate the expressions using Table 5.8 Give exact values if possible; otherwise, make the best possible estimates using left-hand Riemann sums. Table 5.8 $$\begin{array}{c|c|c|c|c|c|c}\hline t & 0.0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 \\\\\hline f(t) & 0.3 & 0.2 & 0.2 & 0.3 & 0.4 & 0.5 \\\\\hline g(t) & 2.0 & 2.9 & 5.1 & 5.1 & 3.9 & 0.8 \\\\\hline\end{array}$$ $$\int_{0}^{0.5} f(t) d t$$

Let \(f(t)=F^{\prime}(t) .\) Write the integral \(\int_{a}^{b} f(t) d t\) and evaluate it using the Fundamental Theorem of Calculus. $$F(t)=3 t^{2}+4 t ; a=2, b=5$$

Explain what is wrong with the statement. For any function, \(\int_{1}^{3} f(x) d x\) is the area between the the graph of \(f\) and the \(x\) -axis on \(1 \leq x \leq 3\)
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