The area of integration forms a right triangle subtending angle from 45 to 90 degree, so the limit for r would be a function of ##\theta##. For a hint, as you sweep the triangle in between those two limiting angles, the projection ##r \sin \theta## is constant.mmont012 said:
Cartesian coordinates are a system of specifying points in a plane using x and y coordinates. Polar coordinates, on the other hand, use a distance (r) and angle (θ) to specify a point in a plane.
To convert from Cartesian to polar coordinates, you can use the following formulas:
r = √(x²+y²) and θ = tan^-1 (y/x).
Alternatively, you can use the Pythagorean theorem and trigonometric identities to calculate the values.
A double integral in polar coordinates is an integral with two variables, r and θ, where the limits of integration are determined by the region of integration in the polar plane. It is used to calculate the volume or area of a region in polar coordinates.
To evaluate a double integral in polar coordinates, you first need to determine the limits of integration for r and θ based on the region of integration. Then, you can use the appropriate formula for the function being integrated and solve the integral using techniques such as substitution or integration by parts.
Double integrals in polar coordinates have many applications in mathematics and physics. They are commonly used in calculating the area or volume of irregular shapes, finding the center of mass and moments of inertia, and solving differential equations in polar coordinates. They also have applications in fields such as engineering, economics, and computer graphics.