Use cylindrical coordinates. Evaluate ∬E z d V, where E is enclosed by the paraboloid

Use cylindrical coordinates. Evaluate ∬E z d V, where ...

Key Concepts

Integration of Functions over Volumes

Integrating a function like z over a given volume involves setting up the triple integral with the proper limits for z, r, and ?, and then integrating sequentially. This process applies the principles of integration to accumulate values throughout the volume, resulting in the overall quantity of interest.

Coordinate Transformation and the Jacobian

When using cylindrical coordinates, the transformation from Cartesian coordinates introduces a Jacobian determinant (r in this case) that accounts for the change in volume element. This factor ensures that the volume is correctly measured during the integration process.

Setting Integration Bounds for Bounded Regions

In problems involving regions enclosed by surfaces, such as a paraboloid and a plane, it is essential to determine how these surfaces intersect. In cylindrical coordinates, this involves finding the limits for r, ?, and z that describe the entire volume. The parity and intersection of the bounding surfaces guide the setup of these limits.

Cylindrical Coordinates

This coordinate system uses the variables r, ?, and z where x = r cos ? and y = r sin ?. It simplifies the evaluation of integrals over regions that are symmetric about the z-axis, such as those involving circular cross-sections, by aligning the coordinate system with the symmetry of the domain.

Triple Integrals

Triple integrals are used to compute quantities, such as volume or mass, by integrating a function over a three-dimensional region. They extend the concept of single and double integrals to three dimensions and involve integrating with respect to three variables.

Transcript

00:01 Okay, so this problem wants you to evaluate the triple integral of z along the region e, where e is enclosed by the paraboloid z equals x -r -fus y squared, and the plane z equals to 4.

00:15 All right, so the first thing we need to do is convert everything into cylindrical coordinates.

00:20 And i'm looking at the integrand.

00:22 Integrant is just z, and that is going to be the same in rectangular and in cylindrical coordinates.

00:30 We leave that the same.

00:32 Dv, however, that equals to r times dz times dr times d theta.

00:42 And looking at the boundaries, we have the paraboloid z equals xx plus y square, and we know that x squared plus y squared equals to r squared.

00:49 So this can be rewritten as z equals to r squared.

00:53 And the plane is equals four, well, that's just z equals to four in cylindrical coordinates.

00:58 So now we've got everything into cylindrical coordinates.

01:02 Now we have to figure out the boundaries.

01:04 So first we have dz, which is the vertical axis in three dimensions.

01:11 And we know by paraboloid that dz goes from r squared to 4, because the plane of 4 is always going to be the, because z equals r squared is an upper paraboloid facing upwards.

01:30 So the z equals 4 is the ceiling for that paraboloid.

01:35 Next we have dr, and we know that dr is the radius, and we can just figure that out by plugging z equals to 4 into our paraboloid to find when at the top of the maximum radius is...

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