Net area and definite integrals Use geometry (not Riemann...

Net area and definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.
$$\int_{-1}^{3} \sqrt{4-(x-1)^{2}} d x$$

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Key Concepts

Definite Integral as Net Area

A definite integral represents the net area between the graph of a function and the x-axis over a specified interval. It sums up positive areas (where the function is above the x-axis) and subtracts areas where the function is below the x-axis, providing a measure of the overall accumulated area. This concept is foundational in calculus and helps in understanding the physical interpretation of integration.

Geometric Interpretation of Integrals

Many integrals can be evaluated by recognizing that the region under the curve corresponds to a familiar geometric shape, which allows the use of area formulas rather than algebraic integration techniques. This approach is particularly useful when the function defines part of a circle, triangle, or other simple shape. It simplifies the process by connecting analytic methods with geometric intuition.

Transformed Circle Equation

The function expressed as sqrt(4 - (x-1)^2) is an example of a transformed circle equation. Specifically, it represents the upper semicircle of a circle with radius 2 centered at (1, 0). Recognizing this form is crucial as it allows one to directly compute the area corresponding to the semicircular shape using the formula for the area of a circle, adjusted for being only a part of it.

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