Area of the region {(x, y)} ∈R^2: y ≥√(|x+3|), 5 y ≤(x+9)...

Area of the region $\{(x, y)\} \in R^{2}: y \geq \sqrt{|x+3|}$,
$5 y \leq(x+9) \leq 15\}$ is equal to $\quad$ [Single Correct Option 2016]
(a) $\frac{1}{6}$
(b) $\frac{4}{3}$
(c) $\frac{3}{2}$
(d) $\frac{5}{3}$

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Key Concepts

Area Computation via Integration

Calculating the area of a specified region typically involves setting up definite or double integrals according to the given boundaries. This concept includes determining the limits of integration by analyzing the inequalities and transforming the region if necessary to facilitate the integration process.

Inequalities in Cartesian Coordinates

Defining regions with inequalities in the xy?plane is fundamental to coordinate geometry. This concept involves interpreting and using inequalities to specify areas bounded by different curves or lines, which then assists in establishing the limits for integration when computing areas.

Absolute Value Functions

Absolute value functions, which result in piecewise definitions, are important in problems where the function behavior changes across different intervals. Understanding how to handle expressions like ?|x+3| is crucial when setting up integration bounds and ensuring the correct evaluation of the area under non-standard curves.

Function Transformations

Transformations such as shifts or scalings are central in simplifying the analysis of complex regions. Recognizing how functions are translated or reflected helps in reformulating integration boundaries into more manageable expressions, thereby streamlining the calculation of areas.

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