Find the centroid (x,y) of the region bounded by y=√x, the x‑axis and the line x=4.

Find the centroid $(\overset{x}{ˉ}, \overset{y}{ˉ})$ of the region bounded by $y = x$ , the $x$ ‑axis and the line $x = 4$ .

hard 4 marks Class 12 · Mathematics · Application of Integrals

Answer

(\frac{12}{5},, \frac{3}{4})

The correct answer is $(\frac{12}{5},, \frac{3}{4})$ — Compute $\overset{x}{ˉ}$ using $, \overset{x}{ˉ} = \frac{1}{A} \int_{0}^{4} x s q r t x, d x$ ; the integral is $\frac{2}{5} x^{5/2}_{0}^{4} = \frac{64}{5}$ , so $\overset{x}{ˉ} = \frac{64/5}{16/3} = \frac{12}{5}$ .

Step-by-step solution

Find the area: $A = \int_{0}^{4} x, d x = \frac{2}{3} x^{3/2}_{0}^{4} = \frac{16}{3}$ .
Compute $\overset{x}{ˉ}$ using $, \overset{x}{ˉ} = \frac{1}{A} \int_{0}^{4} x s q r t x, d x$ ; the integral is $\frac{2}{5} x^{5/2}_{0}^{4} = \frac{64}{5}$ , so $\overset{x}{ˉ} = \frac{64/5}{16/3} = \frac{12}{5}$ .

Common mistake: Forgetting the factor

1/2

when computing

\overset{y}{ˉ}

or mis‑computing the area

Concept tested: Centroid of planar region

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

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Find the centroid (xˉ,yˉ​) of the region bounded by y=x​, the x‑axis and the line x=4.

Step-by-step solution

Find the centroid $(\overset{x}{ˉ}, \overset{y}{ˉ})$ of the region bounded by $y = x$ , the $x$ ‑axis and the line $x = 4$ .