Force of Fluid by Integration: Total hydrostatic force acting on the slant side of trapezoidal trough

This is a basic problem in Hydraulics
$F_H = \gamma \bar{h} A = 62.4(1)(2 \times 6) = 748.8 ~ \text{lb}$

$F_V = \gamma V = 62.4 \times \frac{1}{2}(2)(2)(6) = 748.8 ~ \text{lb}$

$F = \sqrt{{F_H}^2 + {F_V}^2} = 1058.96 ~ \text{lb}$
 

Solution by Integration
$dF = \gamma h \, dA = \gamma (y \sin 45^\circ)(6 \, dy)$

$F = 374.4 \sin 45^\circ {\displaystyle \int_0^{2\sqrt{2}}} y \, dy$

$F = 1058.96 ~ \text{lb}$