Area Under the Curve
MCQ — area of region
Grade 12
Question:
<p>Let \(f(x)\ge0\) be continuous and \(A(t)=\int_0^t f(x)\,dx\). Which hold? [JEE Advanced 2014]</p>
<li>\(A(t)\) is non-decreasing</li>
<li>\(A(0)=0\)</li>
<li>\(A'(t)=f(t)\)</li>
<li>\(A(t)\to\infty\) if \(f\ge\varepsilon>0\)</li>
Step-by-Step Solution
Key Concept: A: A'(t)=f(t)\geq0 \to non-decreasing. B: A(0)=0. C: FTC gives A'=f. D: A(t)\geq\epsilont\to \infty.
<div class='solution'>
<p><strong>A:</strong> $A'(t)=f(t)\ge0\Rightarrow A$ non-decreasing. ✓</p>
<p><strong>B:</strong> $A(0)=\int_0^0 f=0$. ✓</p>
<p><strong>C:</strong> By FTC: $A'(t)=f(t)$. ✓</p>
<p><strong>D:</strong> If $f(t)\ge\varepsilon>0$: $A(t)\ge\varepsilon t\to\infty$. ✓</p>
<p>All four are correct. Answer key: A,D.</p>
</div>
Correct Answer: ['A', 'D']