Find the area enclosed between the curves $y = x^{2}$ and $y = 4 x$ .

medium 2 marks Class 12 · Mathematics · Integrals

Answer

\frac{32}{3}

The correct answer is $\frac{32}{3}$ — Evaluate $2 x^{2} - \frac{x ^{3}}{3}$ from $0$ to $4$ : $(2 \cdot 16 - \frac{64}{3}) - 0 = \frac{32}{3}$ .

Step-by-step solution

Find intersection points: set $x^{2} = 4 x ⟹ x (x - 4) = 0$ , so $x = 0$ or $x = 4$ .
For $0 < x < 4$ , $4 x$ is above $x^{2}$ .
Set up the definite integral: $\int_{0}^{4} (4 x - x^{2}) d x$ .
Integrate term by term: $\int 4 x d x = 2 x^{2}$ and $\int x^{2} d x = \frac{x ^{3}}{3}$ .
Evaluate $2 x^{2} - \frac{x ^{3}}{3}$ from $0$ to $4$ : $(2 \cdot 16 - \frac{64}{3}) - 0 = \frac{32}{3}$ .

Common mistake: Failing to determine which function lies on top or forgetting to subtract the lower function from the upper function

Concept tested: Area between curves

Published 7 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked