Animated Solution for Mathematics - Definite Integration: The area (in sq. units) of the region enclosed between the parabola $y^2=2x$ and the line $x+y=4$ is ____.

The parabola $y^2=2x$ is a horizontal parabola opening towards the positive $x$-axis with its vertex at the origin $(0,0)$.

The line $x+y=4$ acts as a linear boundary, intersecting the axes at $(4,0)$ and $(0,4)$. Together, these curves enclose a specific region in the Cartesian plane.

To determine the boundaries of the region, we find the intersection points by setting the equations equal. We express $x$ in terms of $y$:

Equating these expressions gives:

Multiplying by $2$ results in the quadratic equation:

Factoring the quadratic, we obtain $(y+4)(y-2)=0$, which yields the intersection points at $y=2$ and $y=-4$.

Integrating with respect to $x$ would require splitting the region into two separate integrals. To simplify, we integrate with respect to $y$, treating the area as a sum of horizontal strips.

The area $A$ is defined by the integral of the right-hand curve minus the left-hand curve:

We perform the integration term by term:

Evaluating at the upper limit $y=2$:

Evaluating at the lower limit $y=-4$:

Subtracting the lower limit value from the upper limit value:

The total area enclosed by the curves is 18 square units. Whenever you encounter a horizontal parabola, integrating with respect to $y$ is the most efficient strategy.