Sketch the region whose area is given by the definite integral. Then use a geometric formula to

step 1 In this question we recall that the area under the curve it will be able to express in terms of

Sketch the region whose area is given by the definite...

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Key Concepts

Definite Integrals as Areas

This concept involves interpreting a definite integral as the area under a curve between given boundaries. When the integrand represents a geometric shape, the integral can directly correspond to the area of that shape.

Geometric Representation of Functions

Understanding that the function ?(r² - x²) describes the upper half of a circle (since x² + y² = r²) allows us to visualise the region of integration as a semicircular area. This substitution of calculus with geometric reasoning often simplifies the evaluation of the integral.

Area of a Circle and Its Segments

The total area of a circle of radius r is given by ?r². Consequently, recognizing that the integration limits cover the entire horizontal span of the semicircle (from -r to r), one can determine that the area represented by the integral is exactly half the circle's area, i.e., (?r²)/2.

Sketching Regions Defined by Integrals

Sketching the region corresponding to an integral is an essential technique in calculus that aids in the understanding of the problem. In this case, drawing a semicircle helps in identifying that a geometric formula can be used for finding the area instead of performing the integration through algebraic means.

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