Double Integrals in Polar Coordinates

Transcription

1 Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter suppose the aes are marked in meters. Find the mass of the plate. Solution. As we saw in #b of the worksheet Double Integrals, the mass is the double integral of densit. That is, the mass is + d d. To compute double integrals, we alwas convert them to iterated integrals. In this case, we ll use a double integral in polar coordinates. The region is the polar rectangle θ π, r, so we can rewrite the double integral as an iterated integral in polar coordinates: + d d π/ π/ π/ π/ r cos θ + r sin θr dr dθ r cos θ + sin θ dr dθ r r cos θ + sin θ dθ 7 cos θ + sin θ dθ 7 sin θ cos θ θπ/ θ r. Find the area of the region ling between the curves r + sin θ and r cos θ. You ma leave our answer as an iterated integral in polar coordinates. Solution. As we saw in #a of the worksheet Double Integrals, the area of the region is equal to the double integral d d. To compute the value of this double integral, we will convert it to an iterated integral. This region is not a polar rectangle, so we ll think about slicing. Let s make slices where θ is constant:

2 Our slices go all the wa around the origin, so the outer integral will have θ going from to π. Along each slice, r goes from the inner curve r + sin θ to the outer curve r cos θ. So, the iterated integral is π cos θ +sin θ r dr dθ π r cos θ r r+sin θ π π π dθ [ cos θ + sin θ ] dθ 6 8 cos θ + cos θ sin θ sin θ dθ 8 cos θ + cos 6θ sin θ dθ b the double angle identit cos t cos t sin t θ 8 sin θ + 6 sin 6θ + θπ cos θ π θ. In each part, rewrite the double integral as an iterated integral in polar coordinates. Do not evaluate. a d d where is the left half of the unit disk. Solution. The region is the polar rectangle π θ π, r. In polar coordinates, the integrand is r. So, we can rewrite the double integral as an iterated integral π/ r r dr dθ. π/

3 b d d where is the right half of the ring + 9. Solution. The region is the polar rectangle π θ π, r. In polar coordinates, the integrand is r cos θ. So, we can rewrite the double integral as an iterated integral π/ π/ r cos θ r dr dθ.. ewrite the iterated integral in Cartesian coordinates d d as an iterated integral in polar coordinates. Tr to draw the region of integration. You need not evaluate. Solution. Let s first write the integrand in polar coordinates. Since r cos θ and r sin θ, the integrand can be written as r sin θ cos θ. Net, let s figure out the region of integration. Since the outer integral is something d, we are slicing the interval [, ] on the -ais, so we are making horizontal slices from to. The inner integral tells us that the left side of each slice is on and the right side of each slice is on. is the left half of the circle +, and is the right half of the circle +, so our region of integration with horizontal slices looks like this: This region is the polar rectangle θ π, r. π r sin θ cos θ r dr dθ. So, the integral in polar coordinates is 5. Find the volume of the solid enclosed b the -plane and the paraboloid z 9. You ma leave our answer as an iterated integral in polar coordinates. Normall, we want θ to be between and π. However, if it s more convenient for a polar integral, we rela this restriction.

4 z Solution. Let s break this down into two steps:. First, we ll write a double integral epressing the volume.. Then, we ll convert the double integral to an iterated integral. Notice that the solid can be described as the solid under z 9 over the region, where is where the solid meets the -plane. So, its volume will be 9 d d. Let s describe in more detail. The surface z 9 intersects the -plane z where + 9, so the region is the disk + 9. Now, we ll convert this double integral to an iterated integral. The region is the polar rectangle θ π, r, so we can rewrite the double integral as 9 d d π π π π 8 dθ 8 θπ θ θ 8π 9 r r dr dθ 9r r dr dθ 9 r r r dθ r 6. The region inside the curve r + sin θ and outside the curve r sin θ consists of three pieces. Find the area of one of these pieces. You ma leave our answer as an iterated integral in polar coordinates. Solution. Since we are finding area, our integral will be d d, where is the region of integration. As alwas, to evaluate the double integral, we need to rewrite it as an iterated integral this time, in polar coordinates. Let s make slices where θ constant. When we re dealing with regions that aren t polar rectangles, it s almost alwas easier to slice where θ constant.

5 We are slicing from the θ of the red point to the θ of the blue point. Let s find these. The red point and blue point are points where the curves r + sin θ and r sin θ intersect, so let s solve + sin θ sin θ. This happens when sin θ, or θ π 6, 5π 6. So, the red point has θ π 8, the blue point has θ 5π 8, and our outer integral will have θ going from π 8 to 5π 8. Along a slice, r goes from the inner curve r sin θ to the outer curve r + sin θ, so we can rewrite our double integral as 5π/8 +sin θ 5π/8 r+sin θ r dr dθ dθ π/8 sin θ π/8 r r sin θ 5π/8 π/8 5π/8 [ + sin θ sin θ ] dθ 5 + sin θ dθ π/8 5θ θ5π/8 cos θ 5 5π 9 θπ/8 7. Find the volume of the ice cream cone bounded b the single cone z + and the paraboloid z. z Solution. Let be the projection of the solid onto the -plane; that is, let be the region we see if we look down on the solid from above. This will be a disk, so let s do the integral in polar coordinates. 5

6 First, we ll rewrite everthing in terms of polar coordinates. The cone z + can be rewritten as z r, and the paraboloid z can be rewritten as z r. To find the disk, notice that, if we look at the solid from above, the disk we see is the size of the circle where the two surfaces intersect. The surfaces intersect where r r ; this can be rewritten as r + r, or r + 6r. Since r, the intersection is r. So, the region is a disk centered at the origin with radius. This is a polar rectangle with r, θ < π. One wa to find the volume of the solid is to find the volume under the paraboloid over, find the volume under the cone over, and subtract the latter from the former. That is: volume under volume under z r minus z r equals volume we want over over z z z So, the iterated integral in polar coordinates is π π r r dr dθ r r dr dθ π π π π π r r r dr dθ r r r dr dθ 6 r r dθ r r 7 dθ Notice that we end up simpl integrating the difference between r and r; this is reall the height of the solid above the point r, θ. For an eplanation of wh this works in terms of iemann sums, see #6 of the worksheet Double Integrals. 8. A flat plate is in the shape of the region defined b the inequalities +,,. The densit of the plate at the point, is. Find the mass of the plate. Solution. As we saw in #b of the worksheet Double Integrals, the mass is the double integral of densit. That is, the mass is d d. Here is a picture of the region: This is similar to what ou were asked to do in the homework problem., #. r 6

7 There are four was we could slice: two in Cartesian verticall or horizontall and two in polar where θ is constant or where r is constant. Here are pictures of all four: vertical constant horizontal constant θ constant r constant When slicing verticall, along θ constant, or along r constant, there are multiple tpes of slices. However, if we slice horizontall, there is onl one tpe of slice. This suggests that we should go with horizontal slices. Slicing horizontall amounts to slicing the interval [, ] on the -ais, so the outer integral will be something d. Each slice has its left end on the left edge of the circle + so where and its right end on, so we can rewrite the double integral as d d d d d Find the area of the region which lies inside the circle + but outside the circle +. Solution. Here is a picture of the region, which we ll call : 7

8 There are four was we could slice: two in Cartesian verticall or horizontall and two in polar where θ is constant or where r is constant. Here are pictures of all four: vertical constant horizontal constant θ constant r constant When slicing verticall or horizontall, we can see that there are multiple tpes of slices. When slicing where θ constant or r constant, there is onl one tpe of slice. So, let s do this in polar coordinates. First, let s write the equations of the two circles in polar coordinates. The circle + is just r. The circle + is more complicated: + r cos θ + r sin θ r cos θ + r sin θ r sin θ + r cos θ + sin θ r sin θ r r sin θ r sin θ In the last step, we ve divided both sides b r; this is fine since r > on the circle. We ll use the third picture, where we slice along θ constant. We can use the fourth as well, but we re more used to doing polar integrals b slicing where θ constant. Here s a picture with more detail. Actuall, r at the ver bottom of the circle, but as it s just one point, it doesn t reall matter. 8

9 We are slicing from the θ of the red point to the θ of the blue point. Let s find these values. The red point and blue point are points where the curves r and r sin θ intersect, so let s solve sin θ. This happens when θ π 6 and θ 5π 6. So, the red point has θ π 6, the blue point has θ 5π 6, and our outer integral will have θ going from π 6 to 5π 6. Along each slice, r goes from the lower circle r to the upper circle r sin θ, so the inner integral will have r going from to sin θ. So, we can rewrite our double integral as 5π/6 sin θ π/6 r dr dθ 5π/6 π/6 5π/6 π/6 5π/6 π/6 r sin θ r r sin θ cos θ dθ dθ dθ b the identit sin θ cos θ θ sin θ θ5π/6 π + θπ/6 9