The area (in sq units) of the region {(x, y): y^2 ≥2 x. and...

The area (in sq units) of the region $\left\{(x, y): y^{2} \geq 2 x\right.$ and $\left.x^{2}+y^{2} \leq 4 x, x \geq 0, y \geq 0\right\}$ is $\quad$
(a) $\pi-\frac{4}{3}$
(b) $\pi-\frac{8}{3}$
(c) $\pi-\frac{4 \sqrt{2}}{3}$
(d) $\frac{\pi}{2}-\frac{2 \sqrt{2}}{3}$

Key Concepts

Coordinate Transformation and Integration Techniques

Using coordinate transformations, such as converting to polar coordinates, is a valuable strategy when dealing with circular or radial symmetry in a problem. This method simplifies the integration process by aligning the region's natural boundaries with the coordinate system, thereby facilitating the area computation.

Parabolic Curves

Parabolic curves, which are quadratic in nature, have distinctive geometric properties such as openness in a particular direction. Understanding the orientation and vertex of a parabola, as well as how to rewrite its equation in standard form, is critical when determining the area of regions defined by parabolic inequalities.

Region Determination by Inequalities

Problems involving regions defined by inequalities require a strong grasp of how equations of curves partition the plane. Being able to interpret and sketch the regions defined by inequalities helps in identifying the limits of integration or the subregions whose areas need to be computed.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double?napped cone. In many problems, circles and parabolas (which are specific types of conic sections) appear, and understanding their standard equations and geometric properties is essential for analyzing regions defined by their inequalities.

Circle Geometry

Identifying the properties of a circle through its equation—especially by completing the square—is a fundamental skill. Recognizing the center and radius of a circle allows one to set up proper bounds for integration when calculating areas of regions associated with the circle.

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