3 Problem 4: Consider the ellipse described by (a/a)2 + (y/b)² = 1, with a > b > 0, unequal constants. %3D (a) Set-up the area of the ellipse, using an integral. (b) Solve the integral from (a), to find the area of this ellipse. (c) Suppose we have a second ellipse, (a/b)² + (y/a)2 = 1 (with b and a the same constants). Solve for the points of intersection of these two ellipses. %3D (d) Set-up an integral, or sum of integrals, that computes the area lying inside both ellipses. (Remark: If you could not do (c), let (c, d) be the point of intersection in the first quadrant. Set-up the integral(s) using this.) (c, d) 0. -2 3 -1 0. -3 -2 b= 2 a= 3 Sample figure for Problem 4, where a = and b = The first ellipse is drawn with a solid line; the second with a dashed line. %3D Problem 5: (Extra Credit:) Compute the following integrals. sin" (x) dx where n EN is odd. x| sin(2x)| dx (b) / (1+ cos² x)²
3 Problem 4: Consider the ellipse described by (a/a)2 + (y/b)² = 1, with a > b > 0, unequal constants. %3D (a) Set-up the area of the ellipse, using an integral. (b) Solve the integral from (a), to find the area of this ellipse. (c) Suppose we have a second ellipse, (a/b)² + (y/a)2 = 1 (with b and a the same constants). Solve for the points of intersection of these two ellipses. %3D (d) Set-up an integral, or sum of integrals, that computes the area lying inside both ellipses. (Remark: If you could not do (c), let (c, d) be the point of intersection in the first quadrant. Set-up the integral(s) using this.) (c, d) 0. -2 3 -1 0. -3 -2 b= 2 a= 3 Sample figure for Problem 4, where a = and b = The first ellipse is drawn with a solid line; the second with a dashed line. %3D Problem 5: (Extra Credit:) Compute the following integrals. sin" (x) dx where n EN is odd. x| sin(2x)| dx (b) / (1+ cos² x)²
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:3
Problem 4: Consider the ellipse described by (a/a)2 + (y/b)² = 1, with a > b > 0, unequal
constants.
%3D
(a) Set-up the area of the ellipse, using an integral.
(b) Solve the integral from (a), to find the area of this ellipse.
(c) Suppose we have a second ellipse, (a/b)² + (y/a)2 = 1 (with b and a the same constants).
Solve for the points of intersection of these two ellipses.
%3D
(d) Set-up an integral, or sum of integrals, that computes the area lying inside both ellipses.
(Remark: If you could not do (c), let (c, d) be the point of intersection in the first quadrant.
Set-up the integral(s) using this.)
(c, d)
0.
-2
3
-1
0.
-3 -2
b= 2
a= 3
Sample figure for Problem 4, where a = and b = The first ellipse is drawn with a solid line;
the second with a dashed line.
%3D
Problem 5: (Extra Credit:) Compute the following integrals.
sin" (x) dx where n EN is odd.
x| sin(2x)|
dx
(b) /
(1+ cos² x)²
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