Mathematics 205 HWK 20 Solutions Section 16.3 p755. Problem 5, 16.3, p755. Sketch (or describe) the region of integration for the integral

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1 Mathematics 5 HWK Solutions Section 16. p755 Problem 5, 16., p755. Sketch (or describe) the region of integration for the integral z 1 f(x, y, z) dy dz dx. Solution. For the inner integral both z and x are fixed. Imagine shooting an arrow in the positive y-direction (i.e. parallel to the y-axis, in the direction of increasing y). The limits of integration tell us that the arrow enters the region when y (so when it hits the xz-plane) and leaves the region when y 1 z, so when it hits the right half of the circular cylinder y + z 1, which is a cylinder that encloses the x-axis, having circles of radius 1 as cross sections perpendicular to the x-axis. Then the limits on z and x tell us that y-arrows are needed for every fixed pair of values x and z satisfying 1 z 1 and x 1. So the region in the xz-plane that is needed for the outer two integrals is a rectangle. The conditions on z don t actually limit our half a solid cylinder any further than it s already been limited. The conditions on x tell us to chop off the part of the half-cylinder that lies behind the yz-plane (where x < ) and to chop off the part of the half-cylinder that lines in front of the line x 1. Thus we re left with something that looks like half of a cylindrical log. The log lies to the right of the xz-plane, between the planes x and x 1, and to the left of the semi-cylindrical surface y 1 z. Here s an effort at a sketch. Note: I accidentally did a sketch of the top half of the region, and am not having very good luck with attempts to sketch the full region on screen, so this sketch shows only the top half. This is a quarter of a log. The full region is half of a log. There should be another half below the xy-plane. (See the next page for a computer-generated sketch.) Page 1 of 7 A. Sontag December 1,

2 Math 5 HWK Solns continued Here s a computer-generated sketch (with the axes rotated to afford a better view) showing the four surfaces (x, x 1, y, y 1 z ) that bound the half-log. Problem 15, 16., p755. Find the volume of the pyramid (or tetrahedron) with base in the plane z 6 and sides formed by the three planes y and y x 4 and x + y + z 4. Solution. We need to work out what this tetrahedron is. First think about the planes z 6, y, and y x 4. The first is parallel to the xy-plane, 6 units down. The second, perpendicular to the base plane, is just the xz-plane. The third, also perpendicular to the base plane, is a plane that could be created by sketching the line y x 4 in the xy-plane and then translating it up and down along the z-axis. Let s see what we have so far tells us about the base of the pyramid. Draw an xy-plane and label it z 6. Draw in the two lines y and y x 4. The third side of the base will come from where the fourth and final plane x + y + z 4 meets the plane z 6. On this intersection we will have x + y 1, so sketch that line in as well. See that the base of the pyramid is a triangular region T, say, in the plane z 6. Page of 7 A. Sontag December 1,

3 Math 5 HWK Solns continued Two sides of the tetrahedron (the side where y and the side where y x 4) rise straight up. The third side (where x + y + z 4) rises on a slant until it hits the other two sides, which happens at the point ( 4,,1). Here s a sketch showing the four planes that create the four faces of the tetrahedron. Call the solid region whose volume we wish to find W, say. We need to express the volume integral W 1 dw as a triple iterated integral. If we shoot a z-arrow upward through W, it will enter the region W where z 6 and it will leave the region W where x+y+z 4, or in other words where z 4 x y. Using a little shorthand, we can write the volume of W as z4 x y T z 6 1 dz da. Now we need to set up limits of integration for x and y, so that (x, y) will cover the base region T. It will work best to fix y and shoot an x-arrow. Then the x-arrow enters T where x y 4 and leaves T where x 1 y. The values for y must then vary from y to y 6. Putting all Page of 7 A. Sontag December 1,

4 Math 5 HWK Solns continued these pieces together, we have volume of W 1 y y 4 1 y y 4 4 x y 6 dz dx dy (1 x y) dx dy [ (1 y)x x ] x 1 y xy 4 dy ( 1 (1 y) 1 ) 4 (1 y) (1 y)(y 4) + (y 4) dy ( ) 1 4 (1 y) (1 y)(y 4) + (y 4) dy ( 5 5y + 1 ) 4 y (14y y 4) + (y 8y + 16) dy ( ) 9 4 y 7y + 81 dy [ 4 y 7 y + 81y 16 ] 6 Page 4 of 7 A. Sontag December 1,

5 Math 5 HWK Solns continued Problem 18, 16., p755. Find the average value of the sum of the squares of three numbers x, y, z, where each number is between and. Solution. Let f(x, y, z) x + y + z, and let R denote the brick in -space that is given by the conditions x, y, z. Then the number we want is the average value of f over R. It is easy to compute that the volume of R is 8. We need to find f dv. We can R use an iterated integral, in any one of the 6 possible orders of integration. For instance, R f(x, y, z) dv (x + y + z ) dx dy dz [ ] x x + xy + xz dy dz x ( ) 8 + y + z dy dz [ ] y 8y + y + yz dz y ( ) + 4z dz [ z + 4z ] z Now divide by the volume of R to find that the specified average value is z R f dv volume of R 8 4. I don t know about you, but I found this result a little surprising surprising, that is, that it came out to be a whole number. I was expecting some weird fraction. Note that we could regard this result as saying that if the density at a point (x, y, z) is equal to the square of its distance from the origin, then the average density for this particular brick is 4. Problem 19, 16., p755. Let W be the solid cone bounded by z x + y and z. Decide (without calculating) whether W x + y dv is positive, negative, or zero. Solution. Since the integrand is definitely positive throughout W (except at the very bottom tip point of the cone), the integral will be positive. Page 5 of 7 A. Sontag December 1,

6 Math 5 HWK Solns continued Problem 1, 16., p755. Let W be the solid cone bounded by z x + y and z. Decide (without calculating) whether x dv is positive, negative, or zero. W Solution. This cone has its tip at the origin, and it expands outward and upward until z. The cross sections parallel to the xy-plane are all disks with centers on the z-axis. On the half of the cone that lies in front of the yz-plane, the integrand, x, will be positive. On the other half, which lies behind the yz-plane, the integrand will be negative. Moreover each point in the front half has a matching point in the back half where the integrand takes exactly the opposite value. Therefore the integral over the front half and the integral over the back half will cancel each other out. The total integral will be zero. A shorter description of the key facts would be: the integrand is an odd function of the variable x and the region is symmetric with respect to the plane x. This symmetry makes the total integral zero. Problem, 16., p755. Let W be the solid half-cone bounded by z x + y, z, and the yz-plane with x. Decide whether the integral W x dv is positive, negative, or zero. Solution. Every point of the region of integration satisfies x, and x is the function we are integrating. Thus the integrand is (and it only equals zero on the leftmost face of the region), so the integral is positive. Problem 7, 16., p755. The figure in the text shows part of a spherical ball of radius 5 cm, namely the part that goes from the bottom of the sphere up cm (so the part that lies between a plane just tangent to the bottom of the sphere and a parallel plane cm higher). Write an iterated triple integral that represents the volume of this region. Solution. Call the solid region W. Then we want to write 1 dv as an iterated triple W integral. First we need to introduce a coordinate system. It will probably be simplest to turn W upside down first. Then a coordinate system that puts the top of the sphere at the point (,,5) so the center is at the origin. Then the sphere has the equation x + y + z 5, and the solid W is that portion of the ball x + y + z 5 that lies above the plane z. (See sketch on the next page.) Page 6 of 7 A. Sontag December 1,

7 Math 5 HWK Solns continued It will be easiest to integrate first with respect to z. Fixing (x, y) and shooting a z-arrow upward, we see that the arrow enters the solid where z and leaves it where z hits the sphere, so where z 5 x y. Then the point (x, y) must vary over the disk in the (x, y)-plane that is the shadow of the disk formed by the intersection of the sphere x + y + z 5 with the plane z. This intersection circle has equation x + y or x + y 16. In other words, we need (x, y) to vary over the disk in the xy-plane that lies inside a circle of radius 4 centered at the origin. This gives us either or volume of W volume of W 4 16 x x 16 y 16 y Better yet, take advantage of the symmetry and use, say, volume of W x 5 x y 5 x y 5 x y dz dy dx dz dx dy. dz dy dx Of course this integral is still messy to calculate using rectangular coordinates. You might like to think about how you could find this volume more easily. Page 7 of 7 A. Sontag December 1,