Area Under the Curve
Area of region — all correct
Grade None
Question:
<p>Which of the following are valid representations of areas? [MAU039]</p>
Area under y=x^2 from 0 to 1 = \int_0^1x^2dx
Area under y=\sqrt{x} from 0 to 1 = \int_0^1\sqrt{x} dx
Area between y=x and y=x^2 from 0 to 1 = \int_0^1(x-x^2)dx
Area of unit circle = 4\int_0^\sqrt{1-x^2}dx
Step-by-Step Solution
Key Concept: All four are standard valid area representations using definite integrals.
<div class='solution'>
<p><strong>A:</strong> \(\int_0^1 x^2\,dx=1/3\). ✓</p>
<p><strong>B:</strong> \(\int_0^1\sqrt{x}\,dx=2/3\). ✓</p>
<p><strong>C:</strong> \(\int_0^1(x-x^2)dx=1/6\). ✓ (line above parabola on [0,1])</p>
<p><strong>D:</strong> \(4\int_0^1\sqrt{1-x^2}dx=4\cdot\frac{\pi}{4}=\pi\). ✓ (area of unit circle)</p>
<p>All four are correct. ✓</p>
</div>
Correct Answer: ['A', 'B', 'C', 'D']