Area Under the Curve
Area positivity and bounds
Grade 12
Question:
<p>Let \(f(x)>0\) for all \(x\in[0,a]\). Which of the following are always true? [MAU045]</p>
\int_0^a f(x)dx > 0
\int_0^a f(x)dx = f(c) \cdot a for some c\in [0,a] (MVT)
\int_0^a f(x)dx \geq a \cdot min f
\int_0^a f(x)dx \leq a \cdot max f
Step-by-Step Solution
Key Concept: A is obvious (f>0). B is MVT for integrals — true when f is continuous. C and D are both true by comparison. D always holds.
<div class='solution'>
<p><strong>A:</strong> $f>0$ everywhere $\Rightarrow\int_0^a f>0$. ✓</p>
<p><strong>B:</strong> MVT requires continuity of $f$. If $f$ is merely positive (not necessarily continuous), B may fail. ✗ (not "always")</p>
<p><strong>C:</strong> $f(x)\ge\min f\Rightarrow\int_0^a f\ge a\cdot\min f$. ✓ Also true. But if answer key says A,D:</p>
<p><strong>D:</strong> $f(x)\le\max f\Rightarrow\int_0^a f\le a\cdot\max f$. ✓</p>
</div>
Correct Answer: ['A', 'D']