Use a graphing utility to find the area enclosed by the curves y = x³, y = x, x = 2, and x = 4. Approximate area to two decimal places. A) 56 B) 2 C) 54 D) 17.33 E) 1

Question

Quizplus · Accepted Answer

Using a graphing utility (such as Desmos), we can graph the given curves and find the enclosed area:

From the graph, we can see that the enclosed area is bounded by x = 2 and x = 4, and by the curves y = xS1U1P13S1S1P0 and y = x. We can find the intersections of these curves by setting them equal to each other:

xS1U1P13S1S1P0 = x
xS1U1P13S1S1P0 - x = 0
x(1S1S0S1S1P0) = 0
x = 0 or x = 1

But x = 0 is not within the bounds of x = 2 and x = 4, so we only need to consider the intersection at x = 1. This means we need to split the enclosed area into two parts: one between x = 2 and x = 1, and another between x = 1 and x = 4.

Using the formula for finding the area between two curves, we have:

Area = ∫(2 to 1) [xS1U1P13S1S1P0 - x] dx + ∫(1 to 4) [xS1U1P13S1S1P0 - x] dx
Area = [-1/4(xS1T15S1T2P1 + 2xS1T15S1P1) from 2 to 1] + [1/4(xS1T15S1T2P1 + 2xS1T15S1P1) from 1 to 4]    (integrating and simplifying)
Area ≈ 54.06

Therefore, the best choice is C) 54.

Use a Graphing Utility to Find the Area Enclosed by the Curves