Area Under the Curve
Area between curves
Grade 12
Question:
<p>Let <em>f</em> and <em>g</em> be continuous functions on \(a \leq x \leq b\) and \(p(x) = \max\{f(x), g(x)\}\) and \(q(x) = \min\{f(x), g(x)\}\). The area bounded by the curves \(y = p(x)\), \(y = q(x)\) and the ordinates \(x = a\) and \(x = b\) is given by</p>
<p>(a) \(\displaystyle\int_a^b (f(x) - g(x))\,dx\)</p>
<p>(b) \(\displaystyle\int_a^b (p(x) - q(x))\,dx\)</p>
<p>(c) \(\displaystyle\int_a^b |p(x) - q(x)|\,dx\)</p>
<p>(d) \(\displaystyle\int_a^b |f(x) - g(x)|\,dx\)</p>
Step-by-Step Solution
Key Concept: The area between p(x) = max{f(x),g(x)} and q(x) = min{f(x),g(x)} equals the integral of their difference |f(x) - g(x)|, which simplifies to ∫|f(x) - g(x)|dx regardless of which function is larger in each subinterval.
<p><strong>Step 1:</strong> Recognize that p(x) = max{f(x), g(x)} is the upper envelope and q(x) = min{f(x), g(x)} is the lower envelope at each point x ∈ [a,b].</p><p><strong>Step 2:</strong> The area between these curves is: A = ∫<sub>a</sub><sup>b</sup> [p(x) - q(x)]dx = ∫<sub>a</sub><sup>b</sup> [max{f,g} - min{f,g}]dx</p><p><strong>Step 3:</strong> Observe that max{f,g} - min{f,g} = |f(x) - g(x)| at every point, since one function is always larger and this difference equals the absolute value.</p><p><strong>Step 4:</strong> Therefore: A = ∫<sub>a</sub><sup>b</sup> |f(x) - g(x)|dx, which works whether f > g or g > f in any subinterval, without needing to find intersection points explicitly.</p><p><strong>Step 5:</strong> The area can also be written as ∫<sub>a</sub><sup>b</sup> p(x)dx - ∫<sub>a</sub><sup>b</sup> q(x)dx, representing the difference of integrals of the maximum and minimum functions.</p><p>∴ Answer: B,C,D</p>
Correct Answer: B,C,D