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Area of a region in a plane Let R be a region in a plane that (a, b, c) and boundary C. Let has a unit normal vector n = F = (bz, cx, ay). a. Show that V x F = n. b. Use Stokes' Theorem to show that area of R = F. dr. c. Consider the curve C given by r = (5 sin t, 13 cos t, 12 sin t), for 0 sts 27. Prove that C lies in a plane by showing that r x r' is constant for all t. d. Use part (b) to find the area of the region enclosed by C in part (c). (Hint: Find the unit normal vector that is consistent with the orientation of C.)

Area of a region in a plane Let R be a region in a plane that (a, b, c) and boundary C. Let has a unit normal vector n = F = (bz, cx, ay). a. Show that V x F = n. b. Use Stokes' Theorem to show that area of R = F. dr. c. Consider the curve C given by r = (5 sin t, 13 cos t, 12 sin t), for 0 sts 27. Prove that C lies in a plane by showing that r x r' is constant for all t. d. Use part (b) to find the area of the region enclosed by C in part (c). (Hint: Find the unit normal vector that is consistent with the orientation of C.)

Advanced Engineering Mathematics
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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Area of a region in a plane Let R be a region in a plane that (a, b, c) and boundary C. Let has a unit normal vector n = F = (bz, cx, ay). a. Show that V x F = n. b. Use Stokes' Theorem to show that area of R = F. dr. c. Consider the curve C given by r = (5 sin t, 13 cos t, 12 sin t), for 0 sts 27. Prove that C lies in a plane by showing that r x r' is constant for all t. d. Use part (b) to find the area of the region enclosed by C in part (c). (Hint: Find the unit normal vector that is consistent with the orientation of C.)
Transcribed Image Text:Area of a region in a plane Let R be a region in a plane that (a, b, c) and boundary C. Let has a unit normal vector n = F = (bz, cx, ay). a. Show that V x F = n. b. Use Stokes' Theorem to show that area of R = F. dr. c. Consider the curve C given by r = (5 sin t, 13 cos t, 12 sin t), for 0 sts 27. Prove that C lies in a plane by showing that r x r' is constant for all t. d. Use part (b) to find the area of the region enclosed by C in part (c). (Hint: Find the unit normal vector that is consistent with the orientation of C.)
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