Area Under the Curve
Area Between Two Curves
Grade 12
Question:
<p>Suppose <span style='color:red'>y = f(x)</span> and <span style='color:red'>y = g(x)</span> are two functions whose graphs intersect at the three points <span style='color:red'>(0, 4), (2, 2)</span> and <span style='color:red'>(4, 0)</span> with <span style='color:red'>f(x) > g(x)</span> for <span style='color:red'>0 < x < 2</span> and <span style='color:red'>f(x) < g(x)</span> for <span style='color:red'>2 < x < 4</span>. If <span style='color:red'>∫₀⁴ [f(x) - g(x)]dx = 10</span> and <span style='color:red'>∫₂⁴ [g(x) - f(x)]dx = 5</span>, the area between two curves for <span style='color:red'>0 < x < 2</span> is:</p>
<p>(A) 5</p>
<p>(B) 10</p>
<p>(C) 15</p>
<p>(D) 20</p>
Step-by-Step Solution
Key Concept: Use the relationship between the given integrals to find the area in the first region by partitioning the integral
<p><strong>Step 1:</strong> For <span style='color:red'>0 < x < 2</span>: <span style='color:red'>f(x) > g(x)</span>, so area = <span style='color:red'>∫₀² [f(x) - g(x)]dx</span></p><p><strong>Step 2:</strong> For <span style='color:red'>2 < x < 4</span>: <span style='color:red'>f(x) < g(x)</span>, so area = <span style='color:red'>∫₂⁴ [g(x) - f(x)]dx = 5</span></p><p><strong>Step 3:</strong> We have <span style='color:red'>∫₀⁴ [f(x) - g(x)]dx = ∫₀² [f(x) - g(x)]dx + ∫₂⁴ [f(x) - g(x)]dx = 10</span></p><p><strong>Step 4:</strong> Note that <span style='color:red'>∫₂⁴ [f(x) - g(x)]dx = -∫₂⁴ [g(x) - f(x)]dx = -5</span></p><p><strong>Step 5:</strong> Therefore: <span style='color:red'>∫₀² [f(x) - g(x)]dx - 5 = 10</span></p><p><strong>Step 6:</strong> <span style='color:red'>∫₀² [f(x) - g(x)]dx = 15</span></p><p>∴ Answer is C.</p>
Correct Answer: C