Section 15 3 Double Integrals in Polar Coordinates MATH 251 Fall 2023 In order to compute sR f x y dA we have learned to describe the region R as lying between functions and between numbers Sometimes a region is much easier to describe in polar coordinates than in rectangular coordinates In this section we will learn how to evaluate a double integeral using polar coordinates Polar coordinates were covered in Engineering Mathematics II To review please see Section 10 3 and 10 4 Here is a quick recap De nition of polar coordinates Let P x y be a point on the xy plane The polar coordinates of P is an ordered pair r where r is the distance from O to P and is the angle between the positive x axis and the line OP Conversion between polar and rectangular coordinates 1 x r cos and y r sin 2 r2 x2 y2 and arctan y x Example 1 Given the rectangular coordinates to be 4 3 what is the point s polar coordinates Example 2 Convert the point 2 3 to the rectangular coordinates 1 2 Converting from polar coordinates to rectangular coordinates is straightforward by simply using the formulas x r cos andy r sin However converting from rectangular coordinates to polar coordinates requires more work than nding r and satisfying r2 x2 y2 and arctan y x This is due to convention in de ning the arctan function Here are the steps to convert from rectangular coordinates to polar coordinates 1 Step 1 determing the quadrant of the point P x y 2 Step 2 solve for r satisfying r2 x2 y2 and arctan y x 3 Step 3 if the point is in the rst or the fourth quadrants then use found in Step 2 If the point is in the rst or third quadrants add to found in Step 2 Example 3 Convert the point 1 1 to polar coordinates Example 4 Describe the following region R using polar inequalities second Double integrals in polar coordinates when the region integrated over can be written nicely in polar coordinates Below is the region R r a r b 3 Divide the region into subrectangles Let us compute the shaded subrectangle First we recall the area of the sector below is 1 2r2 1 2 r2 i The shaded subrectangle is the di erence of two sectors having the same angle but di erent radii namely ri 1 and ri respectively Hence its area is i 1 r2 i r2 where r is the average of ri and ri 1 Change to polar coordinates in a double integral if f is continuous on a polar rectangle R given by 0 a r b where 0 2 then ri ri 1 ri ri 1 r r r2 i 1 1 2 1 2 1 2 x R f x y dA Z Z b a f r cos r sin rdrd 4 Example 5 Evaluate the iterated integral by converting to polar coordinates 0 Z p4 x2 Z 2 0 x2 y2 dydx Example 6 Evaluate the iterated integral by converting to polar coordinates 3Z p9 y2 Z 0 p9 y2 x 2y dxdy Example 7 Evaluate sR xdA where R is the region above the x axis bounded by x2 y2 16 the positive x axis and y x 5 If f is continuous on a polar region of the form Then R r h1 r h2 f r cos r sin rdrd x R h1 Z h2 f x y dA Z 0 Z p6x x2 Z 6 0 px2 y2dydx Example 8 Convert the iterated integral to polar coordinates but do not evaluate 6 Similar to what we have seen in Section 15 1 and 15 2 we can use integration in polar coordinates to nd volume of the solid under a surface Example 9 Compute the volume of the solid under z 16 x2 y2 above the xy plane and inside the cylinder x2 y2 9 We can also nd the volume bounded by two di erent surfaces This may require us to nd the intersection of the two surfaces then project it onto the xy plane to have the region of integration Example 10 Find the volume of the solid bounded by the paraboloids z 2 x2 y2 and z x2 y2 7
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