Area Under the Curve
Area between line and parabola
Grade 12
Question:
<p>The area bounded by \(y=x|x|\), \(x\)-axis and the ordinates \(x=-1\) and \(x=1\) is: [MAU003]</p>
<li>\(\dfrac{1}{2}\)</li>
<li>\(\dfrac{1}{3}\)</li>
<li>\(\dfrac{2}{3}\)</li>
<li>\(1\)</li>
Step-by-Step Solution
Key Concept: x|x| = x^2 for x\geq0 and -x^2 for x<0. Area = \int₋_1^0(-x^2-0)... wait, for area use |y|.
<div class='solution'>
<p>\(y=x|x|=\begin{cases}x^2 & x\ge0 \\ -x^2 & x<0\end{cases}\)</p>
<p>Area = \(\int_{-1}^0(-x^2-0)\cdot(-1)\,dx + \int_0^1 x^2\,dx\) (taking absolute values since region below x-axis contributes positive area)</p>
<p>\(= \int_{-1}^0 x^2\,dx + \int_0^1 x^2\,dx = \left[\frac{x^3}{3}\right]_{-1}^0+\left[\frac{x^3}{3}\right]_0^1 = \frac{1}{3}+\frac{1}{3} = \boxed{\frac{2}{3}}\)</p>
</div>
Correct Answer: C