Area Under the Curve
Area Under Curves
nta_abhyas_2025
Grade 12
Question:
The area bounded by the curve $y^2 = 1 - x$ and the lines $y = \frac{lx}{x}$, $x = -1$ and $x = \frac{1}{e}$ is
$\left(\frac{2}{3} - \frac{1}{4}\right)$ sq. units
$\left(\frac{23}{12}\right)$ sq. units
$\left(\frac{23}{12} - \frac{1}{4}\right)$ sq. units
None of these
Step-by-Step Solution
Key Concept: For regions between two curves, integrate the difference of the upper and lower functions over the specified domain.
The area is found by integrating the difference between the upper and lower curves. The region is bounded by $y = \sqrt{1-x^2}$ (semicircle with radius 1) and $y = e^{x+1}$ from $x = -1$ to $x = 0$. Area $= \int_{-1}^{0} (1 - \sqrt{1-x^2})dx + \int_{0}^{1}(1 - e^{x+1})dx = \frac{\pi}{2} - \frac{1}{e}$ sq. units.
Correct Answer: \frac{\pi}{2} - \frac{1}{e}