Application of Integrals: Application Problems (3)

Question

Find the area of the region bounded by the parabola $y^2 = 4x$ and the line $y = 2x - 4$.

Solution — Step by Step

Why This Works

We always have a choice: integrate with respect to $x$ or $y$. Pick the one where the bounding curves are single-valued. For $y^2 = 4x$, $y$ as a function of $x$ is two-valued (upper and lower branches), so integrating in $x$ would require splitting the region. Integrating in $y$ keeps everything single-valued — much cleaner.

Alternative Method

If we insist on integrating in $x$, split at $x = 1$: from $x = 0$ to $x = 1$ the region is bounded above by $y = 2\sqrt{x}$ and below by $y = -2\sqrt{x}$ (parabola only). From $x = 1$ to $x = 4$, bounded above by $y = 2\sqrt{x}$ and below by $y = 2x - 4$. Two integrals to evaluate vs. one — slower but valid.

Common Mistake