Answer to Question #205343 in Calculus for Yanna

Question

Find the centroid of the region bounded by the curves y= x³ and y=4x in the fourth quadrant. Sketch the bounded region.

Answer 1

Here the region is bounded in first and third quadrants.

So, there is no centroid at 4th quadrant.

Centroid at 1st quadrant is,

M_x = (1/2)∫_a^b (f(x)² - g(x)²) dx

= (1/2)((16x³/3) - (x⁷/7))₀¹

= 2.5923

M_y = ∫_a^b x(f(x)-g(x)) dx

= ((4x³/3)-(x⁵/5))$_0^1$

= 3.85714

M = ∫_a^b (f(x)-g(x)) dx

= (2x² - x⁴/4)$_0^1$

= 1.75

Thus, Centroid,(x_i,y_i) = (M_y/M,M_x/M) = (2.20408,1.48289)