Set up an expression to find the area of the shaded region in Figure below

Set up an expression to find the area of the shaded region in Figure below . Then uge your graphing calculator ta evaluate the expression

The Correct Answer and Explanation is:

Step-by-Step General Approach (Textbook Style)

Concept: Area of a Shaded Region Between Curves

To find the area of a shaded region between two curves or under a single curve, the definite integral is used. The area under a curve y=f(x)y = f(x)y=f(x) from x=ax = ax=a to x=bx = bx=b is given by: Area=∫abf(x) dx\text{Area} = \int_a^b f(x)\,dxArea=∫ab​f(x)dx

If there are two curves, where f(x)f(x)f(x) is the upper function and g(x)g(x)g(x) is the lower function, then the area between the curves from x=ax = ax=a to x=bx = bx=b is: Area=∫ab[f(x)−g(x)] dx\text{Area} = \int_a^b [f(x) – g(x)]\,dxArea=∫ab​[f(x)−g(x)]dx

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Example Setup

Suppose the shaded region lies between the curves y=3x−x2y = 3x – x^2y=3x−x2 (on top) and y=xy = xy=x (on the bottom) from x=0x = 0x=0 to x=2x = 2x=2.

The expression for the area becomes: Area=∫02[(3x−x2)−x] dx=∫02(2x−x2) dx\text{Area} = \int_0^2 [(3x – x^2) – x]\,dx = \int_0^2 (2x – x^2)\,dxArea=∫02​[(3x−x2)−x]dx=∫02​(2x−x2)dx

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Evaluating with a Graphing Calculator

1. Enter the function into the Y= screen.
   • In this case: Y1=2x−x2Y_1 = 2x – x^2Y1​=2x−x2
2. Use the fnInt or ∫( function (depending on your calculator model).
   • Example on TI-84:
     • Press MATH → scroll to fnInt(
     • Enter: fnInt(2x - x^2, x, 0, 2)
3. Hit ENTER, and the calculator evaluates the integral.

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Answer to the Example

∫02(2x−x2) dx=[x2−x33]02=(4−83)−(0−0)=43\int_0^2 (2x – x^2)\,dx = \left[ x^2 – \frac{x^3}{3} \right]_0^2 = (4 – \frac{8}{3}) – (0 – 0) = \frac{4}{3}∫02​(2x−x2)dx=[x2−3×3​]02​=(4−38​)−(0−0)=34​

So, the area of the shaded region is 43\frac{4}{3}34​ square units.

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Conclusion

Setting up and evaluating definite integrals allows us to find the exact area of shaded regions under or between curves. The upper and lower functions must be identified clearly from the graph. A graphing calculator simplifies evaluation by numerically computing the integral once the correct expression is input.