Area Under the Curve
Area Between Cubic Curve and Its Tangent
nta_pyq_2023_apr
Grade 12
Question:
The area of the region enclosed by the curve $y=x^3$ and its tangent at the point $(-1,-1)$ is
$\dfrac{19}{4}$
$\dfrac{23}{4}$
$\dfrac{31}{4}$
$\dfrac{27}{4}$
Step-by-Step Solution
Key Concept: Tangent at $(-1,-1)$: slope $=3(-1)^2=3$, equation $y=3x+2$. Find other intersection of tangent with $y=x^3$: $x^3=3x+2\Rightarrow(x+1)^2(x-2)=0$, so $x=2$.
Tangent: $y=3x+2$. Intersection: $x=2$. Area $=\int_{-1}^2(3x+2-x^3)dx=\left[\frac{3x^2}{2}+2x-\frac{x^4}{4}\right]_{-1}^2=\frac{27}{4}$.
Correct Answer: 4