Question

Find the centroid of the region bounded by the graph of the parametric equations and the coordinate axes. $$ x=\sqrt{4-t}, y=\sqrt{t} $$

Short answer

Detailed computations involve integrals that can be evaluated either directly or using numerical methods. The final answer will be the coordinates of the centroid, which are the values of \( \bar{x} \) and \( \bar{y} \).

Step-by-step solution

Identify the Limits for Parameter t

Since \( x = √{4-t}\), is a square root function, it’s domain is \(t ≥ 0\). Also, \( t < 4 \) since \( x \) must be real. Therefore, the range for t is \( 0 ≤ t < 4 \).

Compute Area

Area under the curve is given by \( A = ∫(from 0 to 4) y dx = ∫(from 0 to 4) √t . d(√{4-t}) = ∫(from 0 to 4) √t . (-1/2√{4-t} dt) = ∫(from 0 to 4) -1/2t √{4-t} dt. Now evaluate this integral to find the area A.

Compute x-coordinate of Centroid

The x-coordinate of centroid is given by \( \bar{x} = 1/A ∫(from 0 to 4) x y dx = 1/A ∫(from 0 to 4) √{4-t} √t . d √{4-t} = 1/A ∫(from 0 to 4) √{4-t} . -1/2t √{4-t} dt = 1/A ∫(from 0 to 4) -1/2t (4-t) dt. Now evaluate this integral to find \( \bar{x} \).

Compute y-coordinate of Centroid

The y-coordinate of centroid is given by \( \bar{y} = 1/A ∫(from 0 to 4) y^2 dx = 1/A ∫(from 0 to 4) t d(√{4-t}) = 1/A ∫(from 0 to 4) t . -1/2√{4-t} dt = 1/A ∫(from 0 to 4) -1/2t^2√{4-t} dt. Now evaluate this integral to find \( \bar{y} \).