Concept explainers
Area of a region in a plane Let R be a region in a plane that has a unit normal
a. Show that ▿ × F = n
b. Use Stokes’ Theorem to show that
c. Consider the curve C given by r = 〈5 sin t, 13 cos t, 12 sin t〉, for 0 ≤ t ≤ 2p. Prove that C lies in a plane by showing that
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.)
Want to see the full answer?
Check out a sample textbook solutionChapter 14 Solutions
Calculus: Early Transcendentals (2nd Edition)
Additional Math Textbook Solutions
University Calculus: Early Transcendentals (4th Edition)
Precalculus Enhanced with Graphing Utilities (7th Edition)
Thomas' Calculus: Early Transcendentals (14th Edition)
University Calculus: Early Transcendentals (3rd Edition)
- Evaluate F. dS , where F = and S is the helicoid with S. vector equation r(u, v) upward orientation. 0arrow_forward(+) Find a unit normal vector to the given Surface in Space at the given Point a) Z = excosy +xy3 P:(1.Ae) by hands ) Plot the surface and its normal vector CH Pin the same figure in wx Naximaarrow_forwardplease help mearrow_forwardLet S be the surface defined by the vector function R(u, v) = (u cos v, u – v, u sin v) with u ER and v E (0, 27]. %3D a. Find the equation of the tangent plane to S where (u, v) = (2, E). b. Determine the area of the portion of S where 0arrow_forwardConsider the ray r(t)=o+ td with origin o=(1, 1, 0) and direction d = (1, 1, 1), and the plane N-x= c with unit normal N= (1, 0, 0) and offset c= 10. If the ray intersect with the triangle, where does the ray intersect the plane? (Express the intersection point in coordinates)arrow_forward2. Find a unit normal vector to the surface x +y + z = 5 at the point (0, 1, 2). Ans. ŷ + 2k) V5 (AMIETE, June 2010)arrow_forwardA&B pls. Thank you & I will give thumbs up.arrow_forwardVerify Stokes's theorem over a quarter circular disk with a radius, r = 4 in the 1* F -а, ху-ӑ, 2х (Figure: 1) quadrant for any given vector, B r= 4 A Figure: 1arrow_forwardDetermine the y-coordinate of the centroid of the shaded area. x = ky? I---xarrow_forwardarrow_back_iosarrow_forward_iosRecommended textbooks for you
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning