1. (6 pts each) Compute the following line integrals. Sketch the path of integration and indicate...
6. Calculate the following line integrals of vector fields. Be sure to name any theorems you use; if you don't use a theorem, write "calculated directly2 (d) F . dr, where F(x,y)-(2ry-уг, r2 +3y2-2cy), and C is the piecewise-linear path frorn (1,3) to (5,2 to (12) to (4,1) (e) φ F.dr, where F(z,y)-(3ysin(Zy), 3rsin(2y)+6ry cos(2p)), and C is the ellipse 2 +9y2-64. oriented counter-clockwise 6. Calculate the following line integrals of vector fields. Be sure to name any theorems you...
To evaluate the following integrals carry out these steps. a. Sketch the original region of integration R in the xy-plane and the new region S in the uv-plane using the given change of variables. b. Find the limits of integration for the new integral with respect to u and v. c. Compute the Jacobian. d. Change variables and evaluate the new integral. х x,y): 0 5x57, 7 sys 6 - -x}; use x=7u, y = 6v - u. S5x25x+7y da,...
14. (10 pts) Use Stokes' Theorem to compute F dr where F , y,) 32,5x, -2yand the curve C is given by the triangle with vertices (0,2,0), (3,0,0), and (0,0, 1) with positive orientation 14. (10 pts) Use Stokes' Theorem to compute F dr where F , y,) 32,5x, -2yand the curve C is given by the triangle with vertices (0,2,0), (3,0,0), and (0,0, 1) with positive orientation
No 3 putin uhd e integral lound a r the val- 0 VIIl, 81. EXERCISES Compute the curve integrals of the vector field over the indicated curves. (x,y)=(x2-2xy,y2-2xy) along the, parabola y=x2 from (-2,4) to 2. 0x, y, xz - y) over the line segment from (0,0, 0) to (1, 2, 4), 3, Let r (x2 y2)1/2 Let F(X)-X. Find the integral of F over the circle of radius 2, taken in counterclock wise direction. 4. Let C be a...
2. (a) Sketch the region of integration and evaluate the double integral: T/4 pcos y rsin y dxdy Jo (b) Consider the line integral 1 = ((3y2 + 2mº cos x){ + (6xy – 31sin y)ī) · dr where C is the curve connecting the points (-1/2, 7) and (T1, 7/2) in the cy-plane. i. Show that this line integral is independent of the path. ii. Find the potential function (2, y) and use this to find the value of...
(4) Evaluate the line integral F dr where C is the epicycloid with parametrization given by r(t) 5 cos t - gradient of the function f(x, y) = 3 sin(ry) + cos(y2) cos 5t and y(t) = 5 sin t - sin 5t for 0 < t < 2« and F is the (5) EvaluateF dr where F(x, y) with counterclockwise orientation (2y, xy2and C is the ellipse 4r2 9y2 36 _ F dr where F(r, y) = (x2 -...
Use to fundamental theorem of line integrals to evaluate F dr for 6. F(xy) = (2xy,x2 -y) over the path C from the point (2, 0) to (0, 2) Use to fundamental theorem of line integrals to evaluate F dr for 6. F(xy) = (2xy,x2 -y) over the path C from the point (2, 0) to (0, 2)
Question 22 1 pts Compute the path integral of F = (y,x) along the line segment starting at (1,0) and ending at (3, 1). Question 23 1 pts Consider the vector field F= (1, y). Compute the path integral of this field along the path: start at (0,0) and go up 2 units, then go right 3 units, then go down 4 units and stop. Question 24 1 pts Compute Ss(-y+ye*y)dx + (x + xey)dy, where S is the path:...
1. Compute each of the following integrals using a technique of your choice. Then for each integral identify one other strategy that you could have attempted, and give a brief one- or two-sentence justification of why you chose your approach over the alternative. (a) [4 points] $c F. dr where F(x,y) (3x²e2y + 4ye4r)i + (2x%e24 +e4x – 7)j and C is the curve that runs along the arc y = 1 – x3 from (0,1) to (1, 0), then...
10. Stokes Theorem and Surface Integrals of Vector Fields a Stokes Theorem:J F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sketch of curve C in the xy-plane. Let F(x,y,z) = <2z, x, y, Compute the curl (F) c. d. Find a parametrization of the surface S: G(u,v)ーーーーーーーーーーーーー Compute N(u,v) e. Use Stokes' Theorem to compute Jc F dr 10. Stokes Theorem and Surface...