# 726 Area enclosed by parabola and straigh line | Centroid of Composite Area

**Problem 726**

Locate the centroid of the shaded area enclosed by the curve y^{2} = ax and the straight line shown in Fig. P-726. Hint: Observe that the curve y^{2} = ax relative to the y-axis is of the form y = kx^{2} with respect to the x-axis.

**Solution 726**

$A_1 = \frac{2}{3}(6)(12) = 48 \, \text{ ft}^2$

$x_1 = \frac{3}{5}(12) = 7.2 \, \text{ ft}$

$y_1 = \frac{3}{8}(6) = 2.25 \, \text{ ft}$

Triangular area

$A_2 = \frac{1}{2}(6)(12) = 36 \, \text{ ft}^2$

$x_2 = \frac{2}{3}(12) = 8 \, \text{ ft}$

$y_2 = \frac{1}{3}(6) = 2 \, \text{ ft}$

Shaded area

$A = A_1 - A_2 = 48 - 36$

$A = 12 \, \text{ ft}^2$

$A\bar{x} = A_1x_1 - A_2x_2$

$12\bar{x} = 48(7.2) - 36(8)$

$\bar{x} = 4.8 \, \text{ ft}$ *answer*

$A\bar{y} = A_1y_1 - A_2y_2$

$12\bar{y} = 48(2.25) - 36(2)$

$\bar{y} = 3 \, \text{ ft}$ *answer*