tag16 said:Homework Statement
Evaluate the double integral by converting to polar coordinates.
∫∫ arctan y/x dA; R is the sector in the first quadrant between the circles 1/4= x^2+y^2 and x^2+y^2=1 and the lines y=x/√3 and y=x.
Homework Equations
arctan y/x= θ
The Attempt at a Solution
D={(r,θ) l 1/4≤r≤1, ∏/6≤∏/4}
∫∫ θr drdθ
Which gives me 25∏^2/3072 which is incorrect according to my book. The correct answer is 15∏^2/2304.
tag16 said:I'm not sure, I thought maybe 1/8 but that's not right.
tag16 said:I tried 1/2 when I was originally having difficulty with the problem and still got the wrong answer. I guess I must be missing something else.
Try again, This gives the right answer using the rest of what you have.tag16 said:I tried 1/2 when I was originally having difficulty with the problem and still got the wrong answer. I guess I must be missing something else.
The purpose of converting a double integral to polar coordinates is to simplify the integration process by using a coordinate system that is better suited for certain types of problems. In particular, polar coordinates are useful for solving problems involving circular or symmetric shapes.
The formula for converting a double integral to polar coordinates is:
∫∫R f(x,y) dA = ∫∫D f(r cosθ, r sinθ) r dr dθ
Where r is the radial distance from the origin, θ is the angle with the positive x-axis, R is the region in rectangular coordinates, and D is the corresponding region in polar coordinates.
No, not all double integrals can be converted to polar coordinates. The region of integration must have a circular or symmetric shape for polar coordinates to be applicable. If the region is not circular or symmetric, it may be more appropriate to use rectangular coordinates.
To determine the limits of integration when converting to polar coordinates, you need to consider the boundaries of the region in both rectangular and polar coordinates. The lower and upper limits for the radial coordinate r can be determined by finding the distance from the origin to the outermost and innermost boundaries of the region. The limits for the angular coordinate θ can be determined by considering the angle at which the boundaries intersect with the positive x-axis.
Yes, there is a specific order for converting a double integral to polar coordinates. The first step is to draw the region of integration and identify any circular or symmetric boundaries. Then, determine the limits of integration for r and θ. Next, substitute the polar coordinates into the integrand and convert any differentials using the Jacobian. Finally, evaluate the integral using the new limits of integration.