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:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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