Question

Use cylindrical coordinates to calculate ∫∫∫W f(x,y,z)dV for the given function and region: f(x,y,z)=z,x²+y²≤z≤25

Answer 1

The triple integral ∫∫∫_W f(x,y,z)dV for the function f(x,y,z)=z over the region x²+y²≤z≤25 is calculated using cylindrical coordinates, with limits of integration converted appropriately to this coordinate system. The calculation should be carried out step by step, starting from the innermost integral.

Step-by-step explanation:

To calculate ∫∫∫_W f(x,y,z)dV for the function f(x,y,z)=z over the region x²+y²≤z≤25, we need to use cylindrical coordinates (r, θ, z). Cylindrical coordinates help simplify the triple integral when dealing with symmetrical regions around one of the axes. Here's how to calculate the integral:

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• Convert the limits of integration for the region x²+y²≤z≤25 to cylindrical coordinates. Since x²+y² equals r², the region becomes r²≤z≤25.
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• The lower bound for z is r² and the upper bound is 25.
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• The limits for r are from 0 to √25 and θ ranges from 0 to 2π to cover the entire circular region.
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• Set up the integral as ∫∫µ∫^{2π}_{0} ∫^{25}_{r^2} ∫^{√25}_{0} z r dz dr dθ.
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• Evaluate the integral step by step from the innermost to the outermost integral.

Through this process, we'll find the volume of the region of interest by integrating the function f(x,y,z) = z over the specified limits. Remember that the cylindrical coordinates include an additional factor of r when converting dV to r dz dr dθ.