How do you Evaluate the integral by changing to cylindrical coordinates?

#\int_(-2)^2 \int_(-\sqrt(4-y^(2)))^(\sqrt(4-y^(2)))int_(sqrt(x^2+y^2))^2 (xz) dzdxdy#

Answer 1
#\int_(-2)^2 \int_(-\sqrt(4-y^(2)))^(\sqrt(4-y^(2)))int_(sqrt(x^2+y^2))^2 (xz) \ dz \ dx\ dy#
The limit on the innermost integral defines the volume inside the cone, vertex at origin, concentric with z-axis, radius 2 at #z = 2#.

The other limits lie on or outside this cone and so they can be simplified as follows in cylindrical:

#= \int_(0)^(2pi) \int_0^2 int_(r)^2 underbrace(r cos theta \ z)_(= xz) * underbrace(r \ dz\ dr \ d theta)_(equiv dA)#
#= \int_(0)^(2pi) \int_0^2 [ r^2 cos theta z^2/2 ]_(r)^2 \ dr \ d theta#
#= \int_(0)^(2pi) \int_0^2 2 r^2 cos theta - r^4/2 cos theta \ dr \ d theta#
#= \int_(0)^(2pi) [ 2/3 r^3 cos theta - r^5/10 cos theta ]_0^2 \ d theta#
#=32/15 \int_(0)^(2pi) cos theta \ d theta = 0#
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Answer 2

To evaluate an integral by changing to cylindrical coordinates, follow these steps:

  1. Identify the limits of integration for the new coordinate system.
  2. Express the integrand and the differential volume element in terms of cylindrical coordinates.
  3. Substitute the expressions for the integrand and the differential volume element into the integral.
  4. Evaluate the integral using the limits of integration and the new expressions for the integrand and the differential volume element.
  5. If necessary, convert the result back to the original coordinate system.

Let me know if you need further clarification or if you have a specific integral in mind!

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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