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Integrals in cylindrical coordinates Evaluate the following integrals in cylindrical coordinates. The figures illustrate the region of
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- Set-up a triple integral in cylindrical coordinates to find the volume of the solid above z=4-x^2-y^2 and below z=10-4x^2-4y^2. (Do not evaluate).arrow_forwardA thick spherical shell occupies the region between two spheres of radii a and 2a, both centred on the origin. The shell is made of a material with density p = A(x2 + y2) z2, where A is a constant. Hence, or otherwise, find the mass of the shell by evaluating a suitable volume integral.You may find the substitution u = cosθ useful.arrow_forwardEvaluate the integral: I = ∭s 2x dxdydz where the solid S is defined as: S = { (x, y, z) ∈ R3 : x ≥ 0 ; 0 ≤ y ≤ 2z + 1 ; x2 + y2 + 4z2 ≤ 1 } (a) Describe or sketch the solid S. (b) Evaluate the integral using the shadow method. Show all the workings and explain the methods used.arrow_forward
- Set-up the double integral to find the mass of the surface S : the part of the plane z = 3 − x − 2y in the first octant, if the mass density at any point on the surface is given by δ(x, y, z) = xz with units of mass per unit area. You do not need to evaluate the double integral.arrow_forwardIntegration by parts Evaluate the following integrals using integration by parts. ∫(2w + 4) cos 2w dwarrow_forwardConverting to a polar integral Integrate ƒ(x, y) = [ln (x2 + y2 ) ]/sqrt(x2 + y2) over the region 1<= x2 + y2<= e.arrow_forward
- Work through all integrals. Determine the volumes of the solids of revolution generated by revolving the given region about the given line. Do by the method indicated. - The region bounded by y = sin(x) , y = 0, on [0,pi], is revolved about the y = 1. Do by washers.arrow_forwardThe tetrahedron enclosed by the coordinate planes and the plane 2x + y + z =4 Triple Integrals in Cartesian Coordinates Solve for the boundaries of the tetrahedron and we will keep it to cartesian coordinates. Solve for z, then set z = 0 to solve for y, then set y = 0 to solve for x. By doing so, you will be able to find the boundaries for your triple integral. Set up your triple integral and solve for your answer.arrow_forwardA solid formed when the area between y=2x2 and the x_axis over the interval 0≤x≤2 is rotated about the x_axis . Find a. The volume of the solid of revolution. b. The surface area of the solid of revolution.arrow_forward
- Use cylindrical coordinates.Evaluate the integral, where E is enclosed by the paraboloid z = 2 + x2 + y2, the cylinder x2 + y2 = 7, and the xy-plane. Integrate: ez dVarrow_forwardA. Using polar coordinates, evaluate the improper integral ∫∫R2 e−1(x^2+y^2) dx dy B. Use part A to evaluate the improper integral ∫∞−∞ e−1x^2 dxarrow_forwardy = ex , y = 0 , x = 0 , x= ln2 Draw the region bounded by the curves y = e^x , y = 0 , x = 0 , x= ln2 in the first quartile. Express the area of this region as a double integral. Solve the integral.arrow_forward
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning