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Magazine Chapter 5: Problem 30
True-False Determine whether the statement is true or false. Explain your answer. It is the case that $$ 0<\int_{-1}^{1} \frac{\cos x}{\sqrt{1+x^{2}}} d x $$
Short Answer
Expert verified
The statement is false; the integral equals zero due to symmetry and cancellation.
Step by step solution
01
Analyze the Integrand Function
Consider the function \( f(x) = \frac{\cos x}{\sqrt{1+x^{2}}} \). The cosine function, \( \cos x \), oscillates between -1 and 1, while \( \sqrt{1+x^2} \) is always positive and increases as \( |x| \) increases. Thus, the integrand is a positive function where \( \cos x \geq 0 \) and a negative function where \( \cos x < 0 \).
02
Check the Symmetry of the Function
Notice that the integral is taken symmetrically over the interval \([-1, 1]\). Since \( \cos(-x) = \cos x \), the function is even, meaning \( f(-x) = f(x) \). However, as the expression under the integral is a product, the signs cancel out, making the function symmetrical in the sense of absolute value.
03
Determine the Behavior of the Integrand Over the Interval
Over \([-1, 0]\), the integrand has sections where \( \cos x < 0 \), and over \([0, 1]\), it has \( \cos x > 0 \). The negative and positive values of the cosine lead to cancellation when integrating from \(-1\) to \(1\).
04
Evaluate the Integral Conceptually
Since the function is even and \( \cos x \) symmetrically switches from negative to positive as \( x \) goes from \(-1\) to \(1\), the integral computes to a value very close to zero, as the negative and positive areas cancel each other out over the total interval.
05
Conclude the Evaluation
Thus, the statement that \( 0 < \int_{-1}^{1} \frac{\cos x}{\sqrt{1+x^{2}}} dx \) is actually false, because the integral evaluates to zero due to symmetry and cancellation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integrals
In calculus, definite integrals are used to calculate the net area under a curve within a specified interval. When you see the notation \( \int_{a}^{b} f(x) \, dx \), it implies finding the integral of the function \( f(x) \) from \( x = a \) to \( x = b \). This tells us about the accumulation of quantities, such as distance when given a velocity function.
This concept is vital for analyzing functions that represent physical phenomena. It is useful for understanding how a function behaves between two points.
This concept is vital for analyzing functions that represent physical phenomena. It is useful for understanding how a function behaves between two points.
- If the graph of the function lies entirely above the x-axis within the interval \([a,b]\), the definite integral gives the area under the curve.
- If the graph of the function dips below the x-axis, the integral of those regions will subtract from the total area, reflecting the 'net' area beneath the curve.
- Thus, different sections of the function can contribute positively or negatively to the integral鈥檚 total value, depending on their position relative to the x-axis.
Symmetry in Functions
Symmetry plays an important role in calculus, especially when dealing with integrals. If a function exhibits symmetry over an interval, calculating definite integrals can be simplified.
There are two key types of symmetry:
There are two key types of symmetry:
- Even Symmetry: A function \( f(x) \) is even if \( f(-x) = f(x) \). Visually, this means the function is mirrored across the y-axis. When integrating an even function over a symmetric interval centered at the origin (like \([-a, a]\)), it simplifies to twice the integral from \(0\) to \(a\).
- Odd Symmetry: A function \( g(x) \) is odd if \( g(-x) = -g(x) \). This means each point on one side of the y-axis has a mirror image on the opposite side. When you integrate an odd function over a symmetric interval, the result is zero, because the positive and negative areas cancel each other out.
Even and Odd Functions
Understanding whether a function is even or odd helps in simplifying integration processes. These properties make certain integrals much simpler due to their predictable outcomes over specific intervals.
- Even Functions: Defined by the property \( f(x) = f(-x) \), these functions are symmetric about the y-axis. Examples include \( \cos(x) \) or \( x^2 \). For even functions, integrating over symmetrical intervals often means the effect is doubled, as both sides of the y-axis contribute equally.
- Odd Functions: The characteristic \( f(-x) = -f(x) \) denotes odd functions, such as \( \sin(x) \) or \( x^3 \). Important to note: the integral of an odd function over a symmetric interval like \([-a, a]\) will always result in zero. This is because the areas on either side of the y-axis perfectly cancel each other out.
