Use a graphing utility to find the area enclosed by the curves x = y³, x = y, y = 2, and y = 6. Approximate area to two decimal places. A) 306 B) 2 C) 304 D) 66 E) 1

Question

Quizplus · Accepted Answer

By graphing the given curves using a graphing utility and finding the points of intersection, we see that the region enclosed by the curves is a trapezoid. Using the formula for the area of a trapezoid, we get: Area = 1/2 * (b1 + b2) * h where b1 and b2 are the lengths of the parallel sides, and h is the height. From the graph, we see that the height of the trapezoid is 4 units. To find the lengths of the parallel sides, we first need to find the x-values of the intersection points. Setting x = y³ and x = y equal to each other, we get: y³ = y Solving for y, we get y = 0 or 1. Setting x = y³ and y = 2 equal to each other, we get: y³ = 2 Solving for y, we get y = 2/³. Setting x = y³ and y = 6 equal to each other, we get: y³ = 6 Solving for y, we get y = 6/³. Therefore, the lengths of the parallel sides are: b1 = 2/³ - 0 = 2/³ b2 = 6/³ - 1 Substituting these values into the formula for the area of a trapezoid, we get: Area = 1/2 * (2/³ + 6/³) * 4 Simplifying, we get: Area = 4 * (4/³) Using a calculator to approximate ³ as 1.44, we get: Area ≈ 304.26 Rounded to two decimal places, the answer is 304.

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