The function is given by $y=x|x|$. We can define this piecewise:
$$y = \begin{cases} x^2 & \text{if } x \ge 0 \\ -x^2 & \text{if } x < 0 \end{cases}$$
The area $A$ enclosed between the curve, the x-axis, $x=-2$ and $x=2$ is given by the integral of the absolute value of the function over the interval $[-2, 2]$:
$$A = \int_{-2}^{2} |y| \, dx = \int_{-2}^{2} |x|x|| \, dx$$
Since $|x| \ge 0$, we have $|x|x|| = |x||x| = x^2$.
So, the integral becomes:
$$A = \int_{-2}^{2} x^2 \, dx$$
We can evaluate this integral:
$$A = \left[ \frac{x^3}{3} \right]_{-2}^{2} = \frac{(2)^3}{3} - \frac{(-2)^3}{3}$$
$$A = \frac{8}{3} - \frac{-8}{3} = \frac{8}{3} + \frac{8}{3} = \frac{16}{3}$$
The final answer is $\boxed{\frac{16}{3}}$.
AI Tutor Explanation
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Step-by-Step Solution
**1. Understand the function:**
The function is \(y = x|x|\). We can rewrite this as a piecewise function:
\[
y = \begin{cases}
x^2 & \text{if } x \geq 0 \\
-x^2 & \text{if } x < 0
\end{cases}
\]
**2. Set up the integral:**
We need to find the area enclosed between the curve, the x-axis, and the lines \(x = -2\) and \(x = 2\). Since the function is negative for \(x < 0\), we need to take the absolute value of the integral over that interval. The total area is given by:
\[
\text{Area} = \left| \int_{-2}^{0} -x^2 \, dx \right| + \int_{0}^{2} x^2 \, dx
\]
**4. Calculate the total area:**
\[
\text{Area} = \left| -\frac{8}{3} \right| + \frac{8}{3} = \frac{8}{3} + \frac{8}{3} = \frac{16}{3}
\]
Correct Answer: \(\frac{16}{3}\)
Pedagogical Audit
Bloom's Analysis:
This is an APPLY question because it requires the student to apply their knowledge of integration and absolute value functions to calculate the area under a curve. They must select and implement the correct integration techniques.
Knowledge Dimension:CONCEPTUAL
Justification:The question requires understanding the concept of area under a curve, the properties of absolute value functions, and how to apply integration to find the area. It's not just about recalling a formula but understanding the underlying concepts.
Syllabus Audit:
In the context of CBSE Class 12, this is classified as COMPETENCY. The question assesses the student's ability to apply the concepts of definite integrals to solve a problem, rather than simply recalling a formula from the textbook. It requires understanding the geometric interpretation of the integral and handling the absolute value function.
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