s closed surface The surface limited by 2-62-242 elliptical praboloid ond for Ā =2x+ (1-4x42)+(22-22)E suiface s 2-0 planeer , (22-22)と vektor function on surfaceS calculete the integral l with the Diver gence Theorem.
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- Evaluate the triple integral XYZdV The bounds of the integrals is the the tetrahedron with vertices (0, 0, 0), (0, 1, 0), (1, 1, 0), and (0, 1, 1).A closed rectangular box with faces parallel to the coordinate planes has one bottom corner at the origin and the opposite top corner in the first octant on the plane 5x+6y+z=1. What is the maximum volume of such a box? Volume=Stokes’ Theorem for evaluating line integrals Evaluate theline integral ∮C F ⋅ dr by evaluating the surface integral in Stokes’Theorem with an appropriate choice of S. Assume C has a counterclockwiseorientation. F = ⟨x2 - y2, z2 - x2, y2 - z2⟩; C is the boundary of thesquare | x | ≤ 1, | y | ≤ 1 in the plane z = 0.
- Surface areas Use a surface integral to find the area of the following surfaces. The frustum of the cone z2 = x2 + y2, for 2 ≤ z ≤ 4 (excludingthe bases)Stokes’ Theorem for evaluating line integrals Evaluate theline integral ∮C F ⋅ dr by evaluating the surface integral in Stokes’Theorem with an appropriate choice of S. Assume C has a counterclockwiseorientation. F = ⟨x2 - z2, y, 2xz⟩; C is the boundary of the plane z = 4 - x - y in the first octant.Maximum surface integral Let S be the paraboloidz = a(1 - x2 - y2), for z ≥ 0, where a > 0 is a real number.Let F = ⟨x - y, y + z, z - x⟩. For what value(s) of a (if any)does ∫∫S(∇ x F) ⋅ n dS have its maximum value?
- Stokes’ Theorem for evaluating line integrals Evaluate theline integral ∮C F ⋅ dr by evaluating the surface integral in Stokes’Theorem with an appropriate choice of S. Assume C has a counterclockwiseorientation. F = ⟨2xy sin z, x2 sin z, x2y cos z⟩; C is the boundary of theplane z = 8 - 2x - 4y in the first octant.A parabolic satellite dish is 2 m wide and 1 /2 m deep. Its axis of symmetry is tilted 30 degrees from the vertical. (i) (ii) (iii) (iv) a. Set up, but do not evaluate, a triple integral in rectangular co-ordinates that gives the amount of water the satellite dish will hold. (Hint: Put your coordinate system so that the satellite dish is in “standard position” and the plane of the water level is slanted.) (Caution: The limits of integration are not “nice.”) b. What would be the smallest tilt of the satellite dish so that it holds no waterLet S be the solid of revolution obtained by revolving about the x-axis the bounded region R enclosed by the curve y=e−2x and the lines x=−1, x=1 and y=0. We compute the volume of S using the disk method. a) Let u be a real number in the interval −1≤x≤1. The section x=u of S is a disk. What is the radius and area of the disk? Radius: Area: b) The volume of S is given by the integral (b to a) ∫f(x)dx, where: a= b= and f(x)= c) Find the volume of S. Give your answer with an accuracy of four decimal places. Volume:
- volume of the solid generated when the region bounded by y = 9 − x2 and y = 2x + 6 is revolved about the x-axis.Double integral to line integral Use the flux form of Green’sTheorem to evaluate ∫∫R (2xy + 4y3) dA, where R is the trianglewith vertices (0, 0), (1, 0), and (0, 1).Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. If needed, you can use GeoGebra to graph the region. Also if the evaluation of the double integral is very complicated, you can use the help of your computer - for example you can use wolfram alpha or symbolab. F = (x 2 + y 2) i + (x - y) j; C is the rectangle with vertices at (0, 0), (6, 0), (6, 9), and (0, 9)