Write a triple integral in cylindrical coordinates of the function f(x, y) = x+y over the region bounded by the xy-plane, z = 20 - 3x² - 3y2, and z= x² + y².

Q: Use cylindrical coordinates to calculate the integral of the function f(x, y, z) = z over the region…

A: Given: fx, y, z=z We have to calculate the integral of the given function over the region above the…

Q: Find the centroid of the solid bounded by x² = 4y and y = 5x - x² about the x-axis.

A: We will draw the region and rotate it about x-axis to create a solid of revolution . We will need…

Q: Evaluate the triple integral in rectangular coordinates x=2 y=√√4-x² z=2 L x=-2y=-√4-x² z=√x²+ y²…

A: To evaluate the given triple integral∫x=−2x=2​∫y=−4−x2​y=4−x2​​∫z=x2+y2​z=2​(x2+y2)dzdydxusing…

Q: Determine the solid bounded by the surfaces 9x2+4y2=30 and the plane 9x+4y−6z=0 in the first octant.

A: Given surface is 9x2+4y2=30 and plane is  9x+4y-6z=0 Bounded region in first octane ⇒x≥0, y≥0, z≥0…

Q: Find the centroid of the region bounded by the curves. y=1-x2, y=0

A: We Use the Given Curves Find the Centroid. Firstly We Find Required Area and After we find X and Y…

Q: Find an equation of the tangent plane at P = (0, 3, – 1) to the sur- face with equation ze* + ez+1 =…

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Q: Integrate f(x, y, z) = 6rz over the region in the first octant (x, y, z 20) above the parabolic…

A: Given that f(x,y,z)=6xz . Which can be expressed as f(x,y,z)=6xz=620  (20xz) . And we will find the…

Q: Find the volume of the solid whose base is the region in the xy-plane that is bounded by the…

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Q: Study the analytically of w = L_r(z) at the points z = 0, z = v2,z = -2i

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Q: Find the centroid of a homogeneous half-ball of radius R.

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Q: Prove that the integral of the bottom of a half ellipsoid, (x/a)^2 + (y/b)^2 + (z/c)^2 =1, is equal…

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Q: | Calculate the surface integral /| G(r) dA, where G = (12 the parametrization r(u, v) = [3u, 2v,…

A: Introduction: We know that the formula for the surface integral is given by,…

Q: Evaluate the surface integral. y dS, S is the helicoid with vector equation r(u, v) = (u cos(v), u…

A: Consider the provided question,

Q: Use cylindrical coordinates to evaluate SI vz dv where E is the solid E bounded by the cylinder,…

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Q: Find the centroid (, ) of the region bounded by: =9z²+5x, y=0, and x=8 y 7= BIC

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Q: Converting into cylindrical coordinates the integral r9-(x²+y²) 9-a2 ydzdrde 9-x2

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Q: Find the centroid (, y) of the region bounded by: y = 2x² + 2x, y = 0, and x = 9 y=

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Q: Find the centroid of a lamina enclosed between the circles of radius 1 and 2 centred at the origin,…

A: Consider the diagram of the lamina as per given information, The centriod of the lamina is defined…

Q: Find the area of the portion of the cylindrical surface y = 4 – x2 in the first octant that is…

A: Given : cylindrical surface y=4-x2 in the first octant is between the xy-plane and the surface…

Q: termine the solid bounded by the surfaces 9x2+4y2=30 and the plane 9x+4y−6z=0 in the first octan

A: The given solid is bounded by the surface 9x2+4y2=30 and the plane 9x+4y−6z=0 in the first octant.

Q: There is a curve parametrized by x = 3t, y = 2t°, 0<t an equation for this curve in terms of ne…

A: Here we have,

Q: Find the centroid (x, y) of the region bounded by y = 8x² + 8x, y = 0, x = 0, and x = 4.

A: The region R is bounded by the curves y=8x2+8x,y=0 (the x-axis), x=0 (the y axis), and x=4.