Question

27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are ​formed by​ the graphs of y = 2√x,y = 3−x,and x=3. ​Find the area of each of the regions R₁,R₂, and R₃.

Accepted Answer

Identify the curves and lines that bound each region R₁, R₂, and R₃. From the graph, R₁ is bounded by y = 3 - x and y = 2\sqrt{x}, R₂ is bounded by y = 2\sqrt{x} and y = 3 - x, and R₃ is bounded by y = 2\sqrt{x} and the vertical line x = 3. Find the points of intersection between the curves to determine the limits of integration. Solve for x where y = 2\sqrt{x} equals y = 3 - x by setting 2\sqrt{x} = 3 - x and solving for x. For region R₁, set up the integral for the area between y = 3 - x (upper curve) and y = 2\sqrt{x} (lower curve) from x = 0 to the intersection point found in step 2. The area is given by $$\$$int_0$$^{x_1$} [(3 - x) - 2\sqrt{x}] \, dx. $$For region R₂, set up the integral for the area between y = 2\sqrt{x} (upper curve) and y = 3 - x (lower curve) from the intersection point found in step 2 to x = 3. The area is given by $$\int_{x_1$}^3 [2\sqrt{x} - (3 - x)] \, dx. $$For region R₃, observe that it is bounded by y = 2\sqrt{x} and the vertical line x = 3, and the x-axis (y=0). Set up the integral for the area under y = 2\sqrt{x} from x = 3 to the upper boundary (which is the line x=3, so the vertical boundary). Since the region is a vertical slice, the area is $$\$$int_3$$^{x_2$} 2\sqrt{x} \, dx$$, where $$x_2 is $$the upper limit determined by the graph (likely x=3). Confirm the exact limits from the graph and set up the integral accordingly.

27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.
Find the area of each of the regions R₁,R₂, and R₃.

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Key Concepts

Finding Points of Intersection

Definite Integrals for Area Calculation

Partitioning the Plane into Regions