Contents 15.2 Iterated Integrals..................................... 2 15.3 Double Integrals over General Regions......................... 5 15.4 Double Integrals in Polar Coordinates.......................... 10 15.5 Applications of Double Integrals............................. 11 15.7 The Triple Integral.................................... 12 15.8 Triple Integrals in Cylindrical Coordinates....................... 13 15.9 Triple Integrals in Spherical Coordinates........................ 16 1
15.2 Iterated Integrals If f(x, y) is a function of two variables on R = [a, b] [c, d] we can do partial integration the same way we did partial derivatives. If we look at d c f(x, y) dy then we are integrating y and holding x constant. In fact we get a function of x: A(x) = We can also integrate A(x) with respect to x: d c f(x, y) dy. b A(x) dx = b d a a c f(x, y) dy dx. The integral of f(x, y) over the region R can be expressed as f(x, y) da = b d a c R f(x, y) dy dx. where da = dxdy = dydx. If f(x, y) is continuous then these integrals are interchangeable b d f(x, y) dy dx = d b a c c a f(x, y) dx dy. Example 15.2.1. 4 1 2 1 x 2 + y 2 dy dx. 2
Example 15.2.2. 2 1 0 0 (2x + y) 8 dx dy. Example 15.2.3. 1 2 0 1 xe x y dy dx. 3
Example 15.2.4. Find the volume of the solid bounded by the elliptic paraboloid z = 1 + (x 1) 2 + 4y 2, the planes x = 3 and y = 2 and the coordinate axes. Step 1: DRAW THE REGION 4
15.3 Double Integrals over General Regions What about nonrectangular regions and double integrals? It is mostly the same but now the limits can be functions and not just numbers. There are two types of regions: Type I y y = g 2 (x) a y = g 1 (x) b x Always integrate in the direction of the postive arrow. In this case we will start with y. b y=g2 (x) a y=g 1 (x) f(x, y) dy dx. Type II y b x = g 1 (y) x = g 2 (y) a x Always integrate in the direction of the postive arrow. In this case we will start with x. b x=g2 (y) a x=g 1 (y) f(x, y) dx dy. 5
Example 15.3.1. Sketch the region and evaluate the integral y 2 x=y 2 1 x=y dx dy 3 2 1 1 1 2 3 4 x 1 6
Example 15.3.2. Evaluate 4y da, D = {(x, y) 1 x 2, 0 y 2x} x 3 + 2 D Example 15.3.3. Evaluate x + y da, where A is bounded by y = x and y = x 2. D 7
Example 15.3.4. Find the volume of the solid enclosed by the paraboloid z = x 2 + 3y 2 and the planes x = 0, y = 1, y = x and z = 0 8
Example 15.3.5. Sketch the region and change the order of integration.. 2 4 x 2 0 4 x 2 6x dy dx Example 15.3.6. Sketch the region and change the order of integration. 1 e x 0 1 dy dx 9
15.4 Double Integrals in Polar Coordinates Integrating in polar coordinates When we were integrating f(x, y) da in cartesian coordinates our da was a small rectangle D of size da = dx dy or da = dy dx. Now we want to be able to integrate in polar coordinates so da = r dr dθ. Conversion Factors: x = r cos θ y = r sin θ r 2 = x 2 + y 2 Example 15.4.1. Evaluate 4 y2 and the y-axis. D e x2 y 2 da where D is the region bounded by the semicircle x = 1. sketch the region. 2. convert to polar. 3. set up integral where da = r dr dθ Example 15.4.2. Find the area of the region inside the circle r = 4 cos θ and outside the circle r = 2. 1. sketch the region. 2. set up integral where da = r dr dθ Example 15.4.3. Convert to polar coordinates and integrate. 2 0 1 (x 1) 2 0 x + y dy dx x 2 + y2 10
15.5 Applications of Double Integrals I. Density and Mass and Density = mass volume mass = volume Density In calculus if we know the density ρ(x, y) function we can add up differential masses M = mass = ρ(x, y) da R II. Center of Mass When we calculute the center of mass we need to know the moment about each axis and the mass. We already know how to calculate the mass so M y = xρ(x, y) da and M x = yρ(x, y) da R R Then x = M y M and y = M x M Example 15.5.1. Find the center of mass of the region of constant density bounded by y 2 = 2x and x + y = 4 in the first quadrant. Example 15.5.2. Find the center of mass of a thin plate bounded by y = 1 and y = x 2 when the density is given by ρ(x, y) = y + 1. 11
15.7 The Triple Integral Everything is the same as before but now we are integrating over a ball in space rather than a 2-D region. dv = dx dy dz and Volume = dv Example 15.7.1. Integrating in 3-D 1 π π 0 0 0 R ysinz dx dy dz Example 15.7.2. Let D be the region bounded by the paraboloids z = 8 x 2 y 2 and z = x 2 +y 2. Write iterated integrals for the volume 3 ways. dv = dz dx dy, dv = dz dy dx and dv = dx dy dz Example 15.7.3. Find the volume of the wedge cut from the cylinder x 2 + y 2 = 1 by the planes z = y and z = 0 and z 0. Example 15.7.4. Rewrite 5 ways: 1 1 x 2 1 x 0 0 0 f(x, y, z) dy dz dx 12
15.8 Triple Integrals in Cylindrical Coordinates Recall 2-D: Cartesian: (x, y) Polar: (r, θ) Conversion Factors: x = r cos θ y = r sin θ r 2 = x 2 + y 2 We are going to extend the idea of polar coordinates to 3 dimensions. We do this by considering the xy-plane as a polar coordinate system and the z-axis stays the same. Cylindrical Coordinates: Cartesian: (x, y, z) Cylindrical: (r, θ, z) Conversion Factors from rectangular to cylindrical.: x = r cos θ z = z y = r sin θ r 2 = x 2 + y 2 z (x, y, z) (r, θ, z) y x (r, θ) Example 15.8.1. Translate the equations from the given system to the either rectangular or cylindrical as appropriate. A. z = x 2 + y 2, z 1. B. r = 3 sec θ 13
2-D integration: In cartesian coordinates da = dx dy and in polar coordinates da = r dr dθ. 3-D integration: In rectangular coordinates dv = dx dy dz and in cylindrical coordinates we need to find our dv : dv = r dz dr dθ. Everything else is the same: 1. Draw the region in 3-space. 2. Draw the projection onto the rθ-plane. 3. Choose an order of integration. 4. Establish the limits. Example 15.8.2. Set up the integral for evaluating x2 + y E 2 dv where E is the region that lies inside the cylinder x 2 + y 2 = 16 and between the planes z = 5 and z = 4. Example 15.8.3. Set up the integral for evaluating E ez paraboloid z = 1 + x 2 + y 2, the cylinder x 2 + y 2 = 5 and the xy-plane. dv where E is enclosed by the Example 15.8.4. Set up the integral for the volume of the region between z = x 2 + y 2 and z = 2 x 2 y 2. Example 15.8.5. Evaluate the integral by changing to cylindrical coordinates: 1 1 x 2 2 x 2 y 2 1 1 x 2 x 2 +y 2 (x 2 + y 2 ) 3/2 dz dy dx 14
Find the centroid in cylindrical coordinates: M xy = z dv distance from xy-plane (Cartesian) = z dv distance from xy-plane (Cylindrical) M yz = x dv distance from yz-plane (Cartesian) = r cos θ dv distance from yz-plane (Cylindrical) M xz = y dv distance from xz-plane (Cartesian) = r sin θ dv distance from zz-plane (Cylindrical) x = M yz M y = M xz M z = M xy M 15
15.9 Triple Integrals in Spherical Coordinates We are going to extend the idea of cartesian coordinates (x, y, z) to spherical coordinates where we have a distance from the origin ρ and two angles. One angle is the same as polar coordinates: θ is the angle made from the x-axis. The other angle ϕ is measured from the positive z-axis with 0 ϕ π. Spherical Coordinates: Cartesian: (x, y, z) Cylindrical: (r, θ, z) Spherical: (ρ, θ, ϕ) Conversion Factors from rectangular to cylindrical and spherical: x = r cos θ = ρ sin ϕ cos θ r 2 = x 2 + y 2 y = r sin θ = ρ sin ϕ sin θ ρ 2 = x 2 + y 2 + z 2 = r 2 + z 2 z = z = ρ cos ϕ z ρ (x, y, z) (ρ, θ, ϕ) y x Example 15.9.1. Translate the equations from the given system to the either rectangular or spherical or both as appropriate. A. z = x 2 + y 2, z 1. B. r = 3 sec θ C. tan 2 ϕ = 1 D. 6 cos ϕ 16
3-D integration: In rectangular coordinates: dv = dx dy dz and in cylindrical coordinates: dv = r dz dr dθ. We want dv for spherical coordinates: dv = ρ 2 sin ϕ dρ dϕ dθ. Everything else is the same: 1. Draw the region in 3-space. 2. Draw a ρ arrow to get dρ limits. 3. Figure out how ϕ and θ change. 4. Establish the limits. Example 15.9.2. Set up the integral to find the volume of the smaller region cut from the solid sphere ρ 2 by z = 1. Example 15.9.3. Let D be the region bounded below by z = 0, above by x 2 + y 2 + z 2 = 4 and on the sides by x 2 + y 2 = 1. Set up the integral in spherical coordinates to find the volume. Example 15.9.4. Evaluate the integral by changing to spherical coordinates: 3 9 x 2 3 9 x 2 0 9 x 2 y 2 z x 2 + y 2 + z 2 dz dy dx 17