Question on triple integral polar

In summary, a triple integral in polar coordinates is a mathematical concept used to calculate the volume under a curved surface in three-dimensional space by integrating over three variables: r, θ, and φ. It differs from the triple integral in Cartesian coordinates in terms of the variables used and the form of the integrals. Using polar coordinates can make certain integrals easier to solve and is useful for problems involving circular or spherical symmetry. To convert a triple integral from Cartesian to polar coordinates, the Jacobian determinant must be used. Some real-world applications of triple integrals in polar coordinates include calculating volume and mass in physics and engineering, as well as in fields such as electromagnetism, fluid dynamics, and quantum mechanics.
  • #1
yopy
43
0
Find the mass of a solid bounded by
x = (4-y2)1/2
y = 0
z = 0
z = 1 + x
with density = y

i understand how to set it upand transform to polar and how to do it but my teacher said its supposed to be -pi/2 to pi/2 for the integral with respect to theta. shouldn't it be 0 to pi/2 because its bounded by y = 0?
 
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  • #2
Maybe "y=0" should be "x=0"? The given equations don't seem to make sense without x=0 being included.
 

FAQ: Question on triple integral polar

What is a triple integral in polar coordinates?

A triple integral in polar coordinates is a mathematical concept used to calculate the volume under a curved surface in three-dimensional space. It involves integrating over three variables: r, θ, and φ.

How is the triple integral in polar coordinates different from the triple integral in Cartesian coordinates?

In polar coordinates, the variables are r (distance from the origin), θ (angle from the positive x-axis), and φ (angle from the positive z-axis). In Cartesian coordinates, the variables are x, y, and z. The integrals themselves are also different, as the limits of integration and the integrands are expressed in terms of the polar variables in one case and the Cartesian variables in the other.

What are the benefits of using polar coordinates in a triple integral?

Using polar coordinates in a triple integral can make certain integrals easier to solve, as the limits of integration and the integrand may be simpler in terms of the polar variables. Additionally, polar coordinates are useful when dealing with problems involving circular or spherical symmetry.

How do you convert a triple integral in Cartesian coordinates to polar coordinates?

To convert a triple integral from Cartesian to polar coordinates, you must use the Jacobian determinant to change the variables. The Jacobian determinant for the conversion from Cartesian to polar coordinates is r2sin(φ). Then, you can substitute the new limits of integration and the new integrand in terms of the polar variables.

What are some real-world applications of triple integrals in polar coordinates?

Triple integrals in polar coordinates are commonly used in physics and engineering to calculate the volume and mass of objects with circular or spherical symmetry. They are also used in fields such as electromagnetism, fluid dynamics, and quantum mechanics to solve problems involving three-dimensional shapes.

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