Q.Find the area of the region in the first quadrant bounded by the parabola $y^2 = 4x$, the $y$-axis, and the line $y = 4$.

The region lies in the first quadrant, bounded by the parabola $y^2=4x$, the $y$-axis, and the line $y=4$. Since the boundary on the right is the curve and on the left is the $y$-axis, it is easiest to integrate with respect to $y$. Solving for $x$:

$x = \frac{y^2}{4}.$ …

Solve the parabola's equation for $x$ in terms of $y$, set up a horizontal strip of length $x(y)$, and integra …

Integrating with respect to $x$ instead (harder to set up correctly for this boundary) or mis-simplifying $\tfrac{1}{4}\cdot\ …