Area Under the Curve
Matrix match — area of standard regions
Grade None
Question:
<p>Match the areas: (P) \(\int_0^\pi\sin x\,dx\) | (Q) area between \(y=x\) and \(y=x^3\) on \([0,1]\) | (R) area of \(x^2+y^2=1\). [MAU047]</p>
P=2, Q=1/2, R=\pi
P=2, Q=1/4, R=\pi
P=1, Q=1/4, R=2\pi
P=2, Q=1/3, R=\pi
Step-by-Step Solution
Key Concept: P: [-cosx]_0^\pi=2. Q: \int_0^1(x-x^3)dx=1/2-1/4=1/4. R: \pi(1)^2=\pi.
<div class='solution'>
<p>(P): \(\int_0^\pi\sin x\,dx=[-\cos x]_0^\pi=1+1=2\). ✓</p>
<p>(Q): \(\int_0^1(x-x^3)dx=[\frac{x^2}{2}-\frac{x^4}{4}]_0^1=\frac{1}{2}-\frac{1}{4}=\frac{1}{4}\). ✓</p>
<p>(R): Area of unit circle \(=\pi\). ✓</p>
<p>Match: P=2, Q=1/4, R=π → Option B. ✓</p>
</div>
Correct Answer: ['B']