Area Under the Curve
Area Under Curves
nta_abhyas_2025
Grade 12
Question:
The area bounded by the curve $y = x^2$ and $y = \frac{-2}{(1 + x)}$ is
\left(\pi - \frac{1}{4}\right) sq units
\left(\pi - \frac{2}{3}\right) sq units
\frac{(\pi+1)}{3} sq units
None of these
Step-by-Step Solution
Key Concept: To find the area between a curve and the x-axis, integrate the absolute value of the function over the specified interval.
The required area is $2$ times the area under the curve from $x = -\frac{1}{2}$ to $x = \frac{1}{2}$, where $y \leq 0$ and $y = 2x^3 - x^2$. Computing the integral: $\text{Area} = 2\int_{-1/2}^{1/2} |2x^3 - x^2| dx = 2\int_{-1/2}^{1/2} (2x^3 - x^2) dx$. Evaluating: $2\left[\frac{2x^4}{4} - \frac{x^3}{3}\right]_{-1/2}^{1/2} = 2\left[\frac{x^4}{2} - \frac{x^3}{3}\right]_{-1/2}^{1/2} = \frac{1}{12}$ sq. units.
Correct Answer: 2