Determine the area of the shaded region in the following figures.

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Identify the two curves that bound the shaded region. The upper curve is y = √(1 - x), and the lower curve is y = √(x/2) + 1.

Determine the points of intersection of the two curves by setting √(1 - x) = √(x/2) + 1 and solving for x. This will give the limits of integration.

Set up the integral for the area of the shaded region. The area is given by the integral of the difference between the upper curve and the lower curve: A = ∫[a to b] (√(1 - x) - (√(x/2) + 1)) dx, where a and b are the x-coordinates of the points of intersection.

Simplify the integrand if possible, and split the integral into manageable parts if necessary. For example, you may need to handle the square root terms separately.

Evaluate the integral step by step, applying appropriate integration techniques such as substitution if needed. The result will give the area of the shaded region.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral

A definite integral calculates the net area under a curve between two specified points on the x-axis. It is represented as ∫[a, b] f(x) dx, where f(x) is the function being integrated, and a and b are the limits of integration. This concept is crucial for finding the area of the shaded region between two curves, as it allows us to quantify the space enclosed by the curves.

Area Between Curves

The area between two curves can be found by integrating the difference of the functions that define the curves over a specified interval. If f(x) is the upper curve and g(x) is the lower curve, the area A can be calculated using the formula A = ∫[a, b] (f(x) - g(x)) dx. Understanding this concept is essential for determining the area of the shaded region in the given problem.

Intersection Points

Intersection points of two curves are the x-values where the curves meet, which are critical for setting the limits of integration when calculating the area between them. To find these points, one must solve the equation f(x) = g(x). Identifying these points ensures that the area calculation accurately reflects the region of interest, as it defines the boundaries for the definite integral.

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