4. (a) Let D be the region located in the first quadrant of R2 between the...
(a) Let D be the region located in the first quadrant of R2 between the two circles of radii 1 and 2 centered on the origin. Evaluate 5. dxdy 1(22 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 log (4(1x4 + p(z, у) in grams per units of area. Show that the mass of the disk exceeds 2r log 2...
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...
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
(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
4. Evaluate ſfx da, where D is the region in the first quadrant that lies between = 1 and x + y = 2 D
(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 =...
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...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...