Let C be an ellipse centered at the origin with horizontal axis of length 2a and vertical axis of length 2b, where a, b > 0. (a) Find two distinct unit-speed parameterizations of C. (b) Using one of your parameterizations from part (a), compute the area of the region bounded by C, x = 0 and y = b . (c) Starting with the standard equation for this ellipse in rectangular coordinates, convert to polar coordinates. If possible, find r(θ). (d) Rotate C about the x-axis to generate a solid. Set up, but do not evaluate, an integral to compute the surface area of this solid using i. one of the parameterizations in part (a) ii. polar coordinates iii. rectangular coordinates (e) Which description of C is easiest to use to compute the surface area of the solid in part (d)? Explain. If possible, compute the surface area using one or more of the integrals set up in (d).
Let C be an ellipse centered at the origin with horizontal axis of length 2a and vertical axis of length 2b, where a, b > 0.
(a) Find two distinct unit-speed parameterizations of C.
(b) Using one of your parameterizations from part (a), compute the area of the region bounded by C, x = 0 and y = b .
(c) Starting with the standard equation for this ellipse in rectangular coordinates, convert to polar coordinates. If possible, find r(θ).
(d) Rotate C about the x-axis to generate a solid. Set up, but do not evaluate, an integral to compute the surface area of this solid using
i. one of the parameterizations in part (a) ii. polar coordinates
iii. rectangular coordinates
(e) Which description of C is easiest to use to compute the surface area of the solid in part (d)? Explain. If possible, compute the surface area using one or more of the integrals set up in (d).
Step by step
Solved in 2 steps with 2 images
