Sketch the region enclosed by the given curves. Decide...

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $x$ or $y$. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
$y=x+1, \quad y=9-x^2, \quad x=-1, \quad x=2$

Key Concepts

Determining Boundaries and Intersection Points

Identifying the boundaries of a region is essential in setting up the problem. This includes finding the points where the curves intersect as well as any additional boundary constraints. These intersection points help in defining the limits of integration, which are critical to accurately computing the area.

Area Between Curves

This concept involves finding the area of a region that is bounded by two or more functions. Typically, one function represents the upper boundary and another represents the lower boundary of the region. The area is computed by setting up a definite integral that subtracts the lower function from the upper function over the interval defined by the intersection of the curves or given boundaries.

Choosing the Variable of Integration

Deciding whether to integrate with respect to x or y depends on the orientation and nature of the bounded region. One typically chooses the variable that simplifies the expression of the top and bottom (or right and left) functions, and minimizes the need for splitting the region into multiple parts. This decision directly affects the setup of the integral and the complexity of the calculations.

Approximating Rectangles (Differential Area Element)

The method of slicing a region into thin rectangles is fundamental to integral calculus. Each small rectangle, defined by a differential element (either dx or dy), represents an approximation of the area over a small slice of the region. Integrating these differential elements over the entire range of interest allows for the exact calculation of the total area, embodying the concept of a Riemann sum in the limit.

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