Discussion
Two straight lines pass through the origin \((x_{0}, y_{0}) = (0,0)\). One of them passes through the point \((x_{1}, y_{1}) = (1,3)\) and the other passes through the point \((x_{2}, y_{2}) = (1,2)\).
What is the area enclosed between the straight lines in the interval [0, 1] on the \(x\)-axis?
Answer: A
The enclosed area is 0.5.
The line through (0, 0) and (1, 3) is \(y = 3x\); the line through (0, 0) and (1, 2) is \(y = 2x\). Between \(x = 0\) and \(x = 1\) the first lies above the second, so the area between them is
\[ \begin{aligned} \int_{0}^{1} (3x - 2x)\,dx &= \left[\frac{x^{2}}{2}\right]_{0}^{1} \\ &= 0.5 \end{aligned} \]
Geometrically it is a triangle with vertices (0, 0), (1, 3) and (1, 2): base 1 (the vertical side at \(x = 1\)) and height 1, area \(\frac{1}{2}\). 1.5 and 1.0 are the areas under the individual lines, not between them, and 2.0 matches neither the area between them nor either line.
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