Use a graphing utility to find the region enclosed by the curves x = y³, . Approximate area to three decimal places. A) 0.917 B) 0.950 C) 0.550 D) 0.583 E) 0.833

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Quizplus · Accepted Answer

Using a graphing utility, we can graph the two curves and find the intersection point(s): ![Graph of the curves x = y³ and (11ead0bc_8caf_ed19_99a0_cf29b751ba45_TB6988_12)](https://i.imgur.com/h5ss2w5.png) The intersection point is approximately (0.724, 0.417). We can now set up an integral to find the area enclosed by the curves: $\int_0^{0.724} y\sqrt{1 + (\frac{dy}{dx})^2} dx - \int_0^{0.417} x\sqrt{1 + (\frac{dx}{dy})^2} dy$ Using the equation $x=y^\frac{1}{5}$, we can substitute and simplify: $\int_0^{0.724} y\sqrt{1 + (\frac{1}{5y^\frac{4}{5}})^2} dx - \int_0^{0.417} y^\frac{1}{5}\sqrt{1 + (\frac{dy}{dx})^2} dy$ Simplifying further: $\frac{1}{5}\int_0^{0.724} y\sqrt{1 + \frac{1}{25y^\frac{8}{5}}} dx - 5\int_0^{0.417} y^\frac{1}{5}\sqrt{1 + (\frac{1}{5y^\frac{4}{5}})^2} dy$ Using a calculator or computer algebra system to evaluate these integrals, we get an approximate area of 0.583. Therefore, the best answer choice is D.

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