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Chapter 5: Problem 7

Give a geometrical explanation of why \(\int_{a}^{a} f(x) d x=0\)

Short Answer

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Question: Provide a geometrical explanation as to why the integral of a function f(x) evaluated from a to a is equal to 0. Answer: When the limits of integration are equal, such as in \(\int_{a}^{a} f(x) dx\), there is no width between the two vertical lines at \(x=a\), making the width effectively zero. Since the area of a shape requires both width and height, the area under the curve between \(x=a\) and \(x=a\) will be zero, regardless of the height of the curve. Hence, \(\int_{a}^{a} f(x) dx = 0\) for any continuous function \(f(x)\) and for any \(a\) in its domain.

Step by step solution

01

Understand Integrals

The integral of a function \(f(x)\) over an interval \([a,b]\) represents the accumulation of the signed area under the curve of the function between the limits a and b, above the x-axis. If \(f(x) > 0\), the area is positive; if \(f(x) < 0\), the area is negative.
02

Find the Area Under the Curve

The integral symbol \(\int_{a}^{b} f(x) dx\) denotes the area under the curve \(y=f(x)\) between two points \((a, f(a))\) and \((b, f(b))\). The area is bounded by the function, x-axis, and two vertical lines at \(x=a\) and \(x=b\). The integral is positive if \(f(x) > 0\) (above the x-axis) and negative if \(f(x) < 0\) (below the x-axis).
03

Consider the Same Limit of Integration

When the limits of integration are equal, such as in \(\int_{a}^{a} f(x) dx\), there is no horizontal distance or width between the vertical lines. The area of a geometrical shape must have width and height to have a non-zero value; however, in this case, the width is effectively zero.
04

Geometrical Explanation of the Integral

Since the width of the region under the curve between \(x=a\) and \(x=a\) is zero, there is no area under the curve to be calculated. The height of the curve does not matter, because the width is zero, and the area will ultimately be zero. Hence, \(\int_{a}^{a} f(x) dx = 0\) for any continuous function \(f(x)\) and for any \(a\) in the domain of \(f(x)\).

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Most popular questions from this chapter

Additional integrals Use a change of variables to evaluate the following integrals. $$\int_{-\pi}^{0} \frac{\sin x}{2+\cos x} d x$$

Morphing parabolas The family of parabolas \(y=(1 / a)-x^{2} / a^{3}\) where \(a>0,\) has the property that for \(x \geq 0,\) the \(x\) -intercept is \((a, 0)\) and the \(y\) -intercept is \((0,1 / a) .\) Let \(A(a)\) be the area of the region in the first quadrant bounded by the parabola and the \(x\) -axis. Find \(A(a)\) and determine whether it is an increasing, decreasing, or constant function of \(a\).

Additional integrals Use a change of variables to evaluate the following integrals. $$\int_{1}^{2} \frac{4}{9 x^{2}+6 x+1} d x$$

Multiple substitutions Use two or more substitutions to find the following integrals. $$\int_{0}^{\pi / 2} \frac{\cos \theta \sin \theta}{\sqrt{\cos ^{2} \theta+16}} d \theta(\text {Hint}: \text { Begin with } u=\cos \theta .)$$

Additional integrals Use a change of variables to evaluate the following integrals. $$\int \frac{\csc ^{2} x}{\cot ^{3} x} d x$$
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