20. Describe the curve, surface, or region given in spherical coordinates 2<p < 3,0 <0<27, 034...
Sketch the region given in spherical coordinates by the inequalities 0<p<1, 0<0 < /2, 0 < ¢ < T. Express this region in cylindrical coordinates.
A) solve this integral in cylindrical coordinates. b) set up the integral in spherical coordinates (without solving) 10 points Compute the following triple integral: 1/ 1.32 + plav JD where D is the region given by V x2 + y2 <2<2. Hint: z= V x2 + y2 is a cone.
Use spherical coordinates to calculate the triple integral of f(x, y, z) = y over the region x2 + y2 + z2 < 3, x, y, z < 0. (Use symbolic notation and fractions where needed.) S S lw y DV = help (fractions)
The magnetic field intensity in all of space is given in terms of spherical coordinates: (1 point) The magnetic field intensity in all of space is given in terms of spherical coordinates: A/m. sin θ Use this knowledge in both parts below. (a) Find the current density (in spherical coordinates) at the point P, whose Cartesian coordinates are (z,ys) = (85,-15,-2). ANSWER: At P, J a+ ag+ ap A/m2 (b) Find the net current, I,flowing through the conical surface S...
Calculate the integral over the given region by changing to polar coordinates: f(x, y) = 16xyl, 2² + y² < 49 Answer:
Let C be the closed curve defined by r(t) = costi + sin tj + sin 2tk for 0 <t< 27. (a) (5 pts] Show that this curve C lies on the surface S defined by z = 2xy. (b) (20 pts] By using Stokes' Theorem, evaluate the line integral F. dr C where F(x, y, z) = (y2 + cos x)i + (sin y +22)j + xk
4. Let C be the closed curve defined by r(t) = costi + sin tj + sin 2tk for 0 <t< 27. (a) [5 pts) Show that this curve C lies on the surface S defined by z = 2xy. (b) (20 pts) By using Stokes' Theorem, evaluate the line integral F. dr с where F(x, y, z) = (y2 + cos x)i + (sin y + z2)j + xk
2 Suppose that f(x, y) = - and the region D is given by {(x, y) |1<<3,3 <y < 6}. y D Q Then the double integral of f(x, y) over D is S1,612,)dady
[7] Rectangular coordinates of a point are given. Find the polar coordinates for each point such that r20 and 050<21. Sketches have been provided on the scratchwork page. (-2,-2/3) (8, - 8) (-2, 0) (-24, 7) 7 7
3. Find the length of the curve y = for 0 < I<2.