Please show simple and clear steps for the problem above will leave a rating if so thank you
where R is the region over which the function is required to find area.
Here, ,
Please show simple and clear steps for the problem above will leave a rating if so...
7. Evaluate the following integral by converting to polar coordinates: S], 127 (2x – y)dA, where R is the region in the first quadrant enclosed by the circle x2 + y2 = 4 and the lines x = 0 and y = x. 8. Find the surface area of the portion of the plane 3x + 2y +z = 6 that lies in the first octant. 9. Use Lagrange multipliers to maximize and minimize f(x, y) = 3x + y...
All of 10 questions, please. 1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the cylinder x2 + y2- 16 below z-2x2 + 2y2 and above the xy - plane. 4. Determine the surface area of the portion of 2x + 3y + 6z - 9 that is in the 1st octant. » 5. Evaluate JSxz...
(i) The sides of a given grain silo are represented by the equation of the cylinder x2 +y-3. The top of the silo is the portion of the sphere x2 + y2 + z2-7 lying within the cylinder and above the zy plane. Sketch and find the volume of the silo using an appropriate coordinate system Q2. [10] (ii) Given that C is the boundary of the plane 2x +2y+z = 6 that lies in the first octant and F...
1. Use cylindrical coordinates to SET UP the integral for the volume of the portion of the unit ball, 22 +232 + x2 < 1, above the plane z = 12 2. (a) Write in spherical coordinates the equations of the following surfaces: (i) x2 + y2 + x2 = 4 (ii) z = 3x2 + 3y2 (b) SET UP the integral in spherical coordinates for the volume of the solid inside the surface 22 + y2 + x2 =...
Please answer all parts a,b,c with clear steps, please remember to answer C. A non-uniform surface charge density of 5p/(p? + 1) nC/m2 lies in the plane z = 2 wherever p< 5; Ps = 0 for p > 5. (a) How much electric flux leaves the circular region p< 5, z = 2? (b) How much electric flux crosses the z = 0 plane in the -a, direction? (c) How much electric flux leaves the cylinder p = 3...
EXAMPLE 4 Find the volume of the solid that lies under the paraboloid z 5x2 - 5y2, above the xy-plane, and inside the cylinder x2 + y2-2x (x-1)2 + y2=1 or r 2 cos 8 SOLUTION The solid lies above the disk D whose boundary circle has equation x2 +y2x or, after completing the square, In polar coordinates we have x2 +y Thus the disk D is given by and x-r cos(), so the boundary circle becomes 2r cos(), or...
QUESTION 5 Let the surface S be the portion of the cylinder x2 + y2 4 under z 3 and above the xy-plane Write the parametric representation r(z,0) for the cylinder x2 +y2 4 in term of z (a) and 0 (2 marks) Based on (a), find the magnitude of llr, x rell for the given cylinder (b) (6 marks) 1 1+ (e) Evaluate z dS for the given S (8 marks) Hence, use the divergence theorem to evaluate f,...
Vector Calculus. Please show steps, explain, and do not use calculator. Thank you, will thumbs up! 3. In this problem, let S be the surface defined be the equations: x2 + y2 + z2 = 1 and x2 + y2 < 1/2 (a) (1 point) Find a parametrization of S 0: DR3 where DC R2 (Hint: use spherical coordinates). (b) (2 points) Use part (a) to find the area of S. (c) (1 point) Let F: R3 R3 be the...
If you could write out the solution neatly so I can study from it 8. (a) Find the center of mass of the cylinder {(x, y, z) : x2 +y2 < 1.0 <之-2). given that the density at point (r,y) is ,y, a2 + ). (Observe that 2.2 у = 0 by symmetry, so that no integration is required to find the and coordinates of the center of mass.)
(For 5b, please use the y-axis as the axis of symmetry for the cylinder) 5) a-b Set-up the flux integrals for the given surfaces in the variables indicated. Your final answer should be a scalar- valued double integral. That is, the double integral should does not contain any vector quantities. The differential is given. Do not solve the integrals you setup in a. and b. No work is needed for a-b. a. F(x, y, z) = 5î + 10ủ +...