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Consider the equations x = y? and x = 4y – 2y². (a) Sketch a graph of these two equations on the same coordinate plane. Label your graph and also label significant points (e.g., points of intersection and vertices). You can use a graphing tool (e.g., Desmos, Geogebra, etc.) to help you. Shade the region bounded completely by the two graphs. (b) Using definite integrals, solve for the area of the region bounded by the two equations in two ways. (They should yield the same final answer.) i. By setting up and solving a sum of definite integrals with respect to x, where the sum represents the area of the shaded region. To do this, one can first solve for x in terms of y for each equation so that the different portions of the curves can be re-expressed as functions of x. The shaded region can then be divided into two appropriate parts. ii. By treating the curves as functions of y and integrating with respect to y. You can read about this method in the latter part of this page:

Consider the equations x = y? and x = 4y – 2y². (a) Sketch a graph of these two equations on the same coordinate plane. Label your graph and also label significant points (e.g., points of intersection and vertices). You can use a graphing tool (e.g., Desmos, Geogebra, etc.) to help you. Shade the region bounded completely by the two graphs. (b) Using definite integrals, solve for the area of the region bounded by the two equations in two ways. (They should yield the same final answer.) i. By setting up and solving a sum of definite integrals with respect to x, where the sum represents the area of the shaded region. To do this, one can first solve for x in terms of y for each equation so that the different portions of the curves can be re-expressed as functions of x. The shaded region can then be divided into two appropriate parts. ii. By treating the curves as functions of y and integrating with respect to y. You can read about this method in the latter part of this page:

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter10: Analytic Geometry
Section10.1: The Rectangular Coordinate System
Problem 51E
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Consider the equations x = y? and x = 4y – 2y?. (a) Sketch a graph of these two equations on the same coordinate plane. Label your graph and also label significant points (e.g., points of intersection and vertices). You can use a graphing tool (e.g., Desmos, Geogebra, etc.) to help you. Shade the region bounded completely by the two graphs. (b) Using definite integrals, solve for the area of the region bounded by the two equations in two ways. (They should yield the same final answer.) i. By setting up and solving a sum of definite integrals with respect to x, where the sum represents the area of the shaded region. To do this, one can first solve for x in terms of y for each equation so that the different portions of the curves can be re-expressed as functions of x. The shaded region can then be divided into two appropriate parts. ii. By treating the curves as functions of y and integrating with respect to y. You can read about this method in the latter part of this page:
Transcribed Image Text:Consider the equations x = y? and x = 4y – 2y?. (a) Sketch a graph of these two equations on the same coordinate plane. Label your graph and also label significant points (e.g., points of intersection and vertices). You can use a graphing tool (e.g., Desmos, Geogebra, etc.) to help you. Shade the region bounded completely by the two graphs. (b) Using definite integrals, solve for the area of the region bounded by the two equations in two ways. (They should yield the same final answer.) i. By setting up and solving a sum of definite integrals with respect to x, where the sum represents the area of the shaded region. To do this, one can first solve for x in terms of y for each equation so that the different portions of the curves can be re-expressed as functions of x. The shaded region can then be divided into two appropriate parts. ii. By treating the curves as functions of y and integrating with respect to y. You can read about this method in the latter part of this page:
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