Problem 7 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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, \(\int_{a}^{a} f(x) dx = 0\).