(a) Let D be the region located in the first quadrant of R2 between the two...
4. (a) Let D be the region located in the first quadrant of R2 between the two circles of radii 1 and 4 centered on the origin. Evaluate ((a2 2) dady sin D 5 marks (b) Consider the thin disk centered on the origin in R2 of radius 1. Suppose it is made of a material with mass density function p(г, у) exp 1 x22 in grams per units of area. Show that the mass of the disk does not...
JJ JR 3. Let R be the first-quadrant region bounded by the circles a2 y 4r, 2y10z and the 6y. Use the transformation -2y, 2 y circles a2 +y and r2 + y r2 + y deimegal ll.rdpdrdy to evaluate the i JJ JR 3. Let R be the first-quadrant region bounded by the circles a2 y 4r, 2y10z and the 6y. Use the transformation -2y, 2 y circles a2 +y and r2 + y r2 + y deimegal ll.rdpdrdy...
(3) Let D be the region in the first quadrant between the circles 12 + y y1 and 2. Sketch the region D and find a C transformation T that maps a rectangular region D (where the sides of D are parallel to the coordinate axes) onto D
3. Let D be the region in the first quadrant lying inside the disk x2 +y2 < 4 and under the line y-v 3 x. Consider the double integral I-( y) dA. a. Write I as an iterated integral in the order drdy. b. Write I as an iterated integral in the order dydx c. Write I as an iterated integral in polar coordinates. d. Evaluate I
(Change of Variables I) Let D be the region in the first quadrant between the hyperbolas xy = 4 and xy = 9, and between the lines x = 9y and y = 9x. (a) Compute the area of D. (b) Compute the centroid of D (i.e., the center of mass of D when D has constant mass density). (c) Does the centroid of D lie inside of D? Hint: Use the change of variables u = ry, v =...
4. Evaluate ſfx da, where D is the region in the first quadrant that lies between = 1 and x + y = 2 D
5. (4 points) A metal disk of radius a, thickness d, and conductivity o is located in the xy plane, centered at the origin. There is a time-dependent uniform magnetic field B(t) = B(t)2. Determine the induced current density J(r,t).
Please solve these three questions! (1) Length of graphs a) Let a path C be given by the graph of y g(x), a 3 < b, with a piecewise C1 function g : [a, b - IR. Show that the path integral of a continuous function f: IR2- R over the path C is b) Let g : [a, b] - IR be a piecewise C1 function. The length of the graph of g on (t, g(t)). Show that [a,b]...
Flag question a) Two point charges of 40 C each, located in the x-y plane at (1,1,0) and (-1,-1,0) respectively. to = 8.85 x 10-12 F/m. V = 4RER i) Sketch this configuration in Cartesian coordinates 12 marks] (8 marks] ii) Determine the electric field at a point 3 m on the z-axis. Q [17 marks) iii) Find the electric potential at any point Palong the x-axis. (5 marks] iv) Evaluate Vat x = 1. [2 marks] b) A thin...
Bl A co-axial cable can be treated as two straight cylindrical layers with length of h -6 cm, radii of a lcm and b 4 cm, that is filled with air. Let there be a charge ( +12 pC) on the inner layer and - -12 pC) on the outer layer (as shown in Fig. Bla). Assume each layer has negligible thickness. -Q bt Fig. Bla Fig. Blb (1) Find the electric field intensity E (p) and potential difference in...