Area Under the Curve
Area Under Curves
nta_abhyas_2025
Grade 12
Question:
A farmer $F_1$ has a land in the shape of a triangle with vertices at $P(0, 0)$, $Q(1, 1)$ and $R(2, 0)$. From this land, a neighbouring farmer $F_2$ takes away the region which lies between the line $PQ$ and a curve of the form $y = x^n$ $(n > 1)$. If the area of the region taken away by the farmer $F_2$ is exactly $30\%$ of the area of $\triangle PQR$, then the value of $n$ is
Step-by-Step Solution
Key Concept: Use the power rule for integration and solve the resulting equation for the exponent.
For the curve $y = x^n$ passing through $(1,1)$, the area under the curve from $0$ to $2$ is $\int_0^2 x^n dx = \frac{x^{n+1}}{n+1}|_0^2 = \frac{2^{n+1}}{n+1}$. Setting this equal to $\frac{16}{5}$: $\frac{2^{n+1}}{n+1} = \frac{16}{5}$ gives $5 \cdot 2^{n+1} = 16(n+1)$. Testing $n=4$: $5 \cdot 32 = 160 = 16 \times 10$, so $n+1 = 5$, thus $n = 4$.
Correct Answer: 4