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- Express the triple integral as an iterated integral (explicit bounds) in cylindrical coordinates and rectangular coordinates. ∫∫∫D yz dV where D is the portion of the solid sphere x2+y2+z2≤ 7 for x ≥ 0 and y ≥ 0.arrow_forwardNo integrals Let F = ⟨2z, z, 2y + x⟩, and let S be the hemisphereof radius a with its base in the xy-plane and center at the origin.a. Evaluate ∫∫S (∇ x F) ⋅ n dS by computing ∇ x F and appealing to symmetry.b. Evaluate the line integral using Stokes’ Theorem to check part (a).arrow_forwardGiven the following triple integral in cylindrical coordinates Find the value of the inside integral, middle integral, and outside integral.arrow_forward
- Evaluate Triple Integral H (6 − x2 − y2) dV, where H is the solid hemisphere x2 + y2 + z2 ≤ 16, z ≥ 0. Use spherical coordinatesarrow_forwardUse cylindrical coordinates.Evaluate the integral, where E is enclosed by the paraboloid z = 7 + x2 + y2, the cylinder x2 + y2 = 7, and the xy-plane. ez dV Earrow_forwardCalc 3 Evaluate the integral, where E is the solid that lies within the cylinder x2 + y2 = 9, above the plane z = 0, and below the cone z2 = x2 + y2. Use cylindrical coordinates.arrow_forward
- Evaluate the triple integral in spherical coordinates. f( rho, theta, phi ) = sin phi, over the region 0 less than or equal to theta less than or equal to 2 pi, 0 less than or equal to phi less than or equal to pi /4, 2 less than or equal to rho less than or equal to 3. Evaluate the triple integral in spherical coordinates. f( rho, theta, phi ) = sin phi, over the region 0 less than or equal to theta less than or equal to 2 pi, 0 less than or equal to phi less than or equal to pi /4, 2 less than or equal to rho less than or equal to 3.arrow_forwardIn the image below, the triple integral is bounded above by x^2 + y^2 + z^2 = 4 and below by z = (sqrt(3))(sqrt(x^2 + y^2)). Set-up the triple integral using: a) Rectangular (Cartesian) coordinates. (Do not evaluate) b) Cylindrical coordinates. (Do not evaluate) c) Spherical coordinates. (Evaluate the Integral)arrow_forwardGiven the following triple integral in spherical coordinates Find the value of the inside integral, middle integral, and outside integral.arrow_forward
- Two variables For a solid bounded on the top by a plane z=y+82, on the bottom, by a plane xy and on the sides by a circular cylinder x2+y2 =6724 a)volume of the solid b)If the integral is performed in polar coordinates, determine the variations of "r" and "θ" in the integration regionarrow_forwardSet-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_forwardFlux Integral, Evaluate double integral S of sin(y)*cos(z)i +e^x*cos(z)j+cos(y)*ln(1+x^2)k)·NdS, where S is the sphere x^2+y^2+z^2=1 oriented outwards.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