Definite Integration
Functions Defined by Integrals
Grade 12
Question:
<p>Let <span>∫₀² <i>e</i><sup><i>x</i></sup> <i>dx</i> = <i>I</i></span> and <span><i>f</i>(<i>t</i>) = ∫ₜ² <i>e</i><sup><i>x</i>²</sup> <i>dx</i></span>, <i>t</i> > 0</p>
<p>(A) <i>f</i>(<i>x</i>) is equal to <i>I</i> – ∫₀ᵗ <i>e</i><sup><i>x</i>²</sup> <i>dx</i></p>
<p>(B) <i>f</i>(<i>x</i>) is monotonic decreasing</p>
<p>(C) If area bounded by <i>y</i> = <i>f</i>(<i>x</i>), <i>x</i> = 2, <i>x</i> = 16 and <i>y</i> = <i>c</i> is minimum then <i>c</i> = (Note: partial text)</p>
<p>(D) Equation of normal to curve <i>y</i> = <i>f</i>(<i>x</i>) at <i>x</i> = 3 is <i>y</i> – 1 = 2(<i>x</i> – 3)</p>
Step-by-Step Solution
Key Concept: Use the Fundamental Theorem of Calculus to express integrals with variable limits and differentiate to find monotonicity.
<p><strong>Solution:</strong></p><p><strong>(A)</strong> By the Fundamental Theorem of Calculus: <i>f</i>(<i>t</i>) = ∫ₜ² <i>e</i><sup><i>x</i>²</sup> <i>dx</i> = ∫₀² <i>e</i><sup><i>x</i>²</sup> <i>dx</i> – ∫₀ᵗ <i>e</i><sup><i>x</i>²</sup> <i>dx</i> = <i>I</i> – ∫₀ᵗ <i>e</i><sup><i>x</i>²</sup> <i>dx</i>. ✓</p><p><strong>(B)</strong> <i>f</i>'(<i>t</i>) = –<i>e</i><sup><i>t</i>²</sup> < 0 for all <i>t</i> > 0, so <i>f</i>(<i>t</i>) is monotonic decreasing. ✓</p>
Correct Answer: A, B