- #1
robertjford80
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Homework Statement
do you see how the integral of r is .5?
I don't get how that follows?
Multiple integrals in polar form are a type of integral used to calculate the area, volume, or other quantities of a two- or three-dimensional shape in polar coordinates. This involves using polar coordinates (r and θ) instead of Cartesian coordinates (x and y) to define the shape and limits of integration.
To convert an integral from Cartesian form to polar form, you must first change the limits of integration from rectangular coordinates to polar coordinates. This involves using the equations x = rcosθ and y = rsinθ to express the limits in terms of r and θ. Then, substitute these new limits into the integral and replace any occurrences of x and y with their polar coordinate equivalents. Finally, integrate with respect to r and θ in their respective ranges.
A single integral in polar form involves integrating a function over a one-dimensional interval, while a double integral involves integrating a function over a two-dimensional region. In polar form, a single integral is typically used to calculate the length of a curve or the area of a sector, while a double integral is used to calculate the area or volume of a shape in polar coordinates.
Multiple integrals in polar form have many applications in fields such as physics, engineering, and mathematics. They are commonly used to calculate the mass, center of mass, and moment of inertia of an object with a radial or rotational symmetry. They are also used in calculating electric and magnetic fields, fluid flow in cylindrical or spherical coordinates, and probability distributions in polar coordinates.
There are several techniques for evaluating multiple integrals in polar form, including using symmetry to simplify the integral, changing the order of integration, and using trigonometric identities to simplify the integrand. In some cases, converting to Cartesian coordinates and then evaluating the integral may also be useful. Additionally, numerical methods such as Monte Carlo integration can be used to approximate the value of a multiple integral in polar form.