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Chapter 5: Problem 7

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

Short Answer

Expert verified
Question: Explain geometrically why \(\int_{a}^{a} f(x) d x=0\). Answer: Geometrically, the integral of a function represents the area under the curve. When both the lower and upper limits of the integral are the same (i.e., \(x=a\)), there is no range along the \(x\)-axis for the curve to span, and thus no area under the curve. Consequently, \(\int_{a}^{a} f(x) dx = 0\).

Step by step solution

01

Understand the integral representation

The integral of a function \(f(x)\), denoted as \(\int_{a}^{b} f(x) dx\), represents the area under the curve of \(f(x)\) between the \(x\)-values \(a\) and \(b\). The integral can also be thought of as the limit of a sum of many small areas created by rectangles under the curve.
02

Visualize the curve between two distinct limits

Suppose we have a function \(f(x)\), and we want to find the area under the curve between \(x=a\) and \(x=b\). We can represent this area as \(\int_{a}^{b} f(x) dx\). As long as \(a \neq b\), the curve will span some range along the \(x\)-axis, forming a region with a non-zero area.
03

Analyze the curve between the same limits

Now, let's consider the case of finding the area under the curve between the same limit, \(x=a\). In this case, the integral is represented as \(\int_{a}^{a} f(x) dx\). Since the lower and upper limits are the same, the curve does not span any range along the \(x\)-axis. There is no area under the curve, as the width of the region is zero.
04

Conclude the result

Based on the geometrical representation of integrals, the area under the curve of a function \(f(x)\) between the same limit - in this case, \(x=a\) - is zero. Therefore, \(\int_{a}^{a} f(x) dx = 0\).

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