Worksheet 3.2: Double Integrals in Polar Coordinates

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1 Boise State Math 75 (Ultman) Worksheet 3.: ouble Integrals in Polar Coordinates From the Toolbox (what you need from previous classes): Trig/Calc II: Convert equations in x and y into r and θ, using the change of variable functions x = r cos θ, y = r sin θ. Calc III: Set up and evaluate double integrals in Cartesian coordinates (this includes finding limits of integration); use the limits of integration for a given double integral to sketch the region of integration; be familiar with applications of the double integral. In particular, know how double integrals are used to compute area, volume, and mass. Goals In this worksheet, you will: Compare Cartesian and polar grids and area elements. Find limits of integration for double integrals in polar coordinates. Use change-of-coordinate functions to convert curves and functions from Cartesian coordinates to polar coordinates. Set up and evaluate double integrals in polar coordinates.

2 Boise State Math 75 (Ultman) Worksheet 3.: ouble Integrals in Polar Coordinates 1 Model 1: Grid Curves in Cartesian and Polar Coordinates iagram 1A: Grid Curves in Cartesian Coordinates (x, y ) Grid Curves: x = constant y varies: < y < Grid Curves: y = constant x varies: < x < Cartesian Grid iagram 1B: Grid Curves in Polar Coordinates (r, θ) Grid Curves: r = constant θ varies: θ < π Grid Curves: θ = constant r varies: r < Polar Grid Critical Thinking Questions In this section, you will compare grid curves in Cartesian and polar coordinates. (Q1) Refer to iagrams 1A & 1B: Which coordinate system has a grid made up of lines parallel to the x- & y-axes? Cartesian / Polar (Q) Refer to iagrams 1A & 1B: Which grid curves are circles centered about the origin? Cartesian, x = constant / Cartesian, y = constant / Polar, r = constant / Polar, θ = constant (Q3) Refer to iagrams 1A & 1B: Which grid curves are rays (half-lines) beginning the origin? Cartesian, x = constant / Cartesian, y = constant / Polar, r = constant / Polar, θ = constant (Q4) Refer to iagram 1B: If the angle θ changes with the radius held constant, the resulting polar grid curves are circles centered about the origin / rays beginning at the origin (Q5) Refer to iagram 1B: If the radius r changes with the angle held constant, the resulting polar grid curves are circles centered about the origin / rays beginning at the origin

3 Boise State Math 75 (Ultman) Worksheet 3.: ouble Integrals in Polar Coordinates (Q6) Graph the following curves, and decide whether they follow grid curves for either the Cartesian or polar grid, or neither. (a) y = 5 Cartesian grid / polar grid / neither (b) x = 5 Cartesian grid / polar grid / neither (c) x = Cartesian grid / polar grid / neither (d) x + y = 9 Cartesian grid / polar grid / neither (e) (x 1) + (y 1) = 9 Cartesian grid / polar grid / neither (f) y = x Cartesian grid / polar grid / neither (g) y = x + 1 Cartesian grid / polar grid / neither (h) y = cos x Cartesian grid / polar grid / neither ( Q7) We will say that a polar rectangle is a region bounded by pairs of polar grid curves r = c 1, r = c, θ = k 1, θ = k (for c 1, c, k 1, k constants). Even though these do not look like actual rectangles, we call them rectangles because sides are formed by pairs of grid curves, which meet at right angles. Sketch three different polar rectangles on the polar grids below: ( Q8) In Cartesian coordinates, both coordinates x and y range from to ( < x, y < ). In polar coordinates, there are restrictions on the coordinates: The radius is non-negative: r <. The angle is restricted to a range of π, for example: θ < π, or π θ < π. Why these restrictions are necessary? Choose a point P = (x, y) that is not the origin. (a) Suppose the angle is restricted to a range of π radians, but the radius is allowed to take on negative values: < r <. How many ways are there to represent your point P in polar coordinates? one way / two ways / infinitely many ways (b) Suppose the radius is restricted to non-negative values r <, but the angle is allowed to take on all values: < θ <. How many ways are there to represent your point P in polar coordinates? one way / two ways / infinitely many ways continued...

4 Boise State Math 75 (Ultman) Worksheet 3.: ouble Integrals in Polar Coordinates 3 (c) Based on your answers to parts (a) and (b), explain why the radius needs to be restricted to positive values, and the angle needs to be restricted to a range of π radians: r < and ω θ < ω + π. Interlude: The Polar Area Element d A Recall from Trigonometry: The length s of an arc of a circle of radius r, subtended by an angle θ, is: s = radius angle = r θ The polar area element da is the area of an infinitesimally small polar rectangle, with sides formed by infinitesimally small segments of radial lines and arcs of circles. The length of the radial line segments is dr (a small change in the r-direction). The length ds of the arcs of circles can be found using the formula ds = radius angle. If the radius is r and the angle is dθ, then: ds = da under the infinite magnifying glass. And the polar area element da is: da = dr ds = dr ( ) = dr dθ = dθ dr (Ask me why I call polar integration, Pirate integration.)

5 Boise State Math 75 (Ultman) Worksheet 3.: ouble Integrals in Polar Coordinates 4 Model : Finding Limits of Integration in Polar Coordinates iagram A r 3 θ π f (r, θ) da = ˆ π ˆ 3 f (r, θ) r dr dθ iagram B r 3 θ π/ f (r, θ) da = ˆ π ˆ 3 f (r, θ) r dr dθ iagram C r π/4 θ π/ f (r, θ) da = ˆ π π 4 ˆ f (r, θ) r dr dθ iagram r / sin θ π/4 θ π/ f (r, θ) da = ˆ π π 4 ˆ sin θ f (r, θ) r dr dθ

6 Boise State Math 75 (Ultman) Worksheet 3.: ouble Integrals in Polar Coordinates 5 Critical Thinking Questions In this section, you will find limits of integration in polar coordinates. (Q9) In iagrams A & B: Write the equation of the boundary circle x + y = 9 in polar coordinates: r = In both iagrams A & B, this corresponds to the upper limit of integration with respect to: r / θ. (Q1) In iagrams B & C: Write the equation of the boundary circle x + y = 4 in polar coordinates: r = In iagram B, this corresponds to the upper / lower limit of integration with respect to r. In iagram C, this corresponds to the upper / lower limit of integration with respect to r. (Q11) In iagram : Use the change-of-coordinate function y = r sin θ to write the equation of the boundary line y = in polar coordinates: r = In iagram, this corresponds to the upper / lower limit of integration with respect to r. (Q1) In all three iagrams A, C &, the lower limit of integration with respect to r is r =. This is because all three of these regions include / do not include the origin. (Q13) In both iagrams A & B, the lower limit of integration with respect to θ is θ =. This is because both of these regions are bounded by the positive x / y -axis. In iagram A, the upper limit of integration with respect to θ is θ = region is bounded by the positive / negative x-axis. In iagram B, the upper limit of integration with respect to θ is θ = region is bounded by the positive / negative x / y -axis., because the, because the (Q14) iagrams C & have the same / different limits of integration with respect to θ. The lower limit of integration with respect to θ for both of these regions is θ =, because both regions are bounded by the line y = x, which makes an angle of with the positive x-axis. The upper limit of integration with respect to θ for both of these regions is θ =, because both regions are bounded by the positive / negative x / y -axis.

7 Boise State Math 75 (Ultman) Worksheet 3.: ouble Integrals in Polar Coordinates 6 (Q15) Let be the disk bounded by the circle x + y = R. (a) Set up a double integral in Cartesian coordinates that gives the area of. (b) Set up a double integral in polar coordinates that gives the area of. (c) Evaluate one of your integrals (your choice) from part (a) or (b) to show that the area of is A = πr ( Q16) Use a double integral in polar coordinates to show that the volume of the sphere x +y +z = 9 is V = 36π.

8 Boise State Math 75 (Ultman) Worksheet 3.: ouble Integrals in Polar Coordinates 7 ( Q17) Convert the integral ˆ 1 ˆ 1 y limits of integration to sketch the region. x + y dx dy to polar coordinates. (Hint: First, use the ( Q18) Suppose a population of squirrels is distributed throughout a region according to the population density σ(x, y) = e (x +y ) (units of squirrels/km ). (a) What does the integral e (x +y ) da represent? (b) Find the total number of squirrels living within 1 km of the central point (, ). ( Q19) is the quarter-disk x + y 9 between the lines y = x and y = x, for x >. Use polar coordinates to evaluate the integral arctan(y/x) da.