(i) Sketch the region bounded by 0 ≤y ≤x^2+1,0 ≤y ≤x+1,0 ≤x ≤2 (ii) Find its area. (iii) Find th

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(i) Sketch the region bounded by 0 ≤y ≤x^2+1,0 ≤y ≤x+1,0 ≤x...

(i) Sketch the region bounded by $0 \leq \mathrm{y} \leq \mathrm{x}^{2}+1,0 \leq \mathrm{y} \leq \mathrm{x}+1,0 \leq \mathrm{x} \leq 2$
(ii) Find its area.
(iii) Find the area of the region bounded byx $^{2}=y-1, x-y+1=0$ and $x=2$ in first quadrant.

Key Concepts

Order of Integration and Setting Limits

Choosing the appropriate order of integration—whether to integrate with respect to x first or y first—is determined by the geometry of the region. A proper selection simplifies the integral setup by providing clear limits of integration based on the functions describing the boundary curves.

Sketching Bounded Regions

Visualizing regions bounded by curves starts with accurately plotting each curve on the coordinate plane and shading the common region where all inequalities are satisfied. This process helps in understanding the spatial relationship between the curves and in identifying the correct integration limits for further analysis.

Intersection of Curves

Determining the intersection points of curves is essential to identify the precise boundaries of the region. By solving the equations simultaneously, one finds the coordinates where the curves meet, which then serve as the limits for integration when calculating the area.

Definite Integration for Area

The area of a region bounded by curves is typically calculated by setting up a definite integral. This involves integrating the difference between the 'top' and 'bottom' functions (or right and left functions, depending on the order of integration) over the interval determined by the intersections of the curves.

First Quadrant Considerations

When a problem specifies the first quadrant, the analysis focuses on regions where both x and y are non-negative. This constraint simplifies the limits for integration and ensures that the region of interest is confined to values that are practically defined, influencing both the sketching and the computation steps.

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