Use cylindrical coordinates. Find the mass and center of mass of the solid S bounded by the paraboloid z = 4x2 + 4y2 and the plane z = a (a > 0) if S has constant density K.
> The total production P of a certain product depends on the amount L of labor used and the amount K of capital investment. In Sections 14.1 and 14.3 we discussed how the Cobb-Douglas model P = bLaK1-a follows from certain economic assumptions, where b and
> Find the extreme values of f on the region described by the inequality. f(x, y) = 2x² + 3y² – 4x – 5, x² + y² < 16
> Find the extreme values of f subject to both constraints. f(x, y, z) = yz + xy; xy= 1, y² + z? = 1 %3D
> Find the extreme values of f subject to both constraints. f (x, y, z) = z; x2 + y2 = x2, x + y + = 24
> Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x1, x2, ..., xn) = x1 + x2 + .. xỉ + x + ... +
> Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y) = xe'; x² + y² = 2
> Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y, z, t) = x + y + z + t; x? + y² + z² + t²
> At a hydroelectric generating station (once operated by the Katahdin Paper Company) in Millinocket, Maine, water is piped from a dam to the power station. The rate at which the water flows through the pipe varies, depending on external conditions. The po
> For this project we locate a rectangular trash Dumpster in order to study its shape and construction. We then attempt to determine the dimensions of a container of similar design that minimize construction cost. 1. First locate a trash Dumpster in your a
> Many rockets, such as the Pegasus XL currently used to launch satellites and the Saturn V that first put men on the moon, are designed to use three stages in their ascent into space. A large first stage initially propels the rocket until its fuel is cons
> Suppose that a solid ball (a marble), a hollow ball (a squash ball), a solid cylinder (a steel bar), and a hollow cylinder (a lead pipe) roll down a slope. Which of these objects reaches the bottom first? (Make a guess before proceeding.) To answer this
> In this project we find formulas for the volume enclosed by a hypersphere in n-dimensional space. 1. Use a double integral and trigonometric substitution, together with Formula 64 in the Table of Integrals, to find the area of a circle with radius r. 2.
> The figure shows the solid enclosed by three circular cylinders with the same diameter that intersect at right angles. In this project we compute its volume and determine how its shape changes if the cylinders have different diameters. 1. Sketch carefu
> Evaluate the iterated integral. ∫_0^1 ∫_0^1 ∫_0^(√(1-z^2 ))z/(y+1) dx dz dy
> Evaluate the iterated integral. ∫_1^2 ∫_0^2x ∫_0^lnxxe^(-y) dy dx dz
> If f is continuous, show that ∫_0^x ∫_0^y ∫_0^z f(t) dt dz dy =1/2 ∫_0^x (x-t)^2 f(t)dt
> Change from rectangular to spherical coordinates. (a). (1, 0, 3 ) (b). ( 3 , -1, - 3 )
> Change from rectangular to spherical coordinates. (a). (0, -2, 0) (b). (-1, 1, - 2 )
> Use spherical coordinates. Find the average distance from a point in a ball of radius a to its center.
> Use spherical coordinates. Evaluate ∭E √(x^2+y^2+z^2 ) dV, where E lies above the cone z = √(x^2+y^2 ) and between the spheres x2 + y2 + z2 − 1 and x2 + y2 + z2 = 4.
> Use spherical coordinates. Evaluate ∭E xe^(x^2+y^2+z^2 ) dV, where E is the portion of the unit ball x2 + y2 + z2 < 1 that lies in the first octant
> Set up the triple integral of an arbitrary continuous function f (x, y, z) in cylindrical or spherical coordinates over the solid shown. ZA y 2.
> Plot the point whose spherical coordinates are given. Then find the rectangular coordinates of the point. (a). (2, π/2, π/2) (b). (4, -π/4, π/3)
> Set up the triple integral of an arbitrary continuous function f (x, y, z) in cylindrical or spherical coordinates over the solid shown. ZA 3- 2 X. y
> Sketch the solid whose volume is given by the integral and evaluate the integral. ∫_0^(π/4) ∫_0^2π ∫_0^secφρ^2 sin φ dρ d θ dφ
> Sketch the solid whose volume is given by the integral and evaluate the integral. ∫_0^(π/6) ∫_0^(π/2) ∫_0^3 ρ^2 sin φ dρ d θ dφ
> (a). Show that when Laplace’s equation (∂^2 u)/(∂x^2 ) + (∂^2 u)/(∂y^2 ) + (∂^2 u)/(∂z^2 ) = 0 is written in cylindrical coordin
> Evaluate the triple integral. ∭E ez/y dV, where E = {(x, y, z) | 0 < y < 1, y < x < 1, 0 < z < xy}
> Plot the point whose spherical coordinates are given. Then find the rectangular coordinates of the point. (a). (6, π/3, π /6) (b). (3, π /2, 3 π/4)
> Evaluate the triple integral. ∭E y dV, where E = {(x, y, z) | 0 < x < 3, 0 < y < x, x - y < z < x + y}
> Evaluate the iterated integral. ∫_0^1 ∫_0^1 ∫_0^(2-x^2 -y^2)〖xye〗^z dz dy dx
> Identify the surface whose equation is given. r2 + z2 = 4
> Set up, but do not evaluate, integral expressions for (a) the mass, (b) the center of mass, and (c) the moment of inertia about the z-axis. The hemisphere x2 + y2 + z2 < 1, z > 0; ρ (x, y, z) = √(x^2 +y^2+z^2 )
> Set up, but do not evaluate, integral expressions for (a) the mass, (b) the center of mass, and (c) the moment of inertia about the z-axis. The solid of Exercise 21; ρ (x, y, z) = √(x^2 +y^2 ) Exercise 21: Use a triple integral to find the volume of
> Assume that the solid has constant density k. Find the moments of inertia for a rectangular brick with dimensions a, b, and c and mass M if the center of the brick is situated at the origin and the edges are parallel to the coordinate axes.
> Assume that the solid has constant density k. Find the moments of inertia for a cube with side length L if one vertex is located at the origin and three edges lie along the coordinate axes.
> Find the mass and center of mass of the solid E with the given density function ρ . E is the tetrahedron bounded by the planes x = 0, y = 0, z = 0, x + y + z = 1; ρ (x, y, z) = y
> The double integral ∫_0^1 ∫_0^11/(1-xy) dx dy is an improper integral and could be defined as the limit of double integrals over the rectangle [0, t] × [0, t] as t → 1-. But if we expan
> Find the mass and center of mass of the solid E with the given density function ρ . E is bounded by the parabolic cylinder z = 1 - y2 and the planes x + z = 1, x = 0, and z = 0; ρ (x, y, z) = 4
> Evaluate the iterated integral. ∫_0^1 ∫_0^2y ∫_0^(x+y)6xy dz dx dy
> Find the mass and center of mass of the solid E with the given density function ρ . E lies above the xy-plane and below the paraboloid z = 1 - x2 - y2; ρ (x, y, z) = 3
> Write five other iterated integrals that are equal to the given iterated integral. ∫_0^1 ∫_y^1 ∫_0^y f (x,y,z) dz dx dy
> Express the double integral in terms of a single integral with respect to r. Then use your calculator to evaluate the integral correct to four decimal places. ∬D √(xy&1+x^2+y^2 ) dA, where D is the portion of the disk x2 + y2 < 1 that lies in the first q
> Express the integral ∭E f (x, y, z) dV as an iterated integral in six different ways, where E is the solid bounded by the given surfaces. x = 2, y = 2, z = 0, x + y - 2z = 2
> Express the integral ∭E f (x, y, z) dV as an iterated integral in six different ways, where E is the solid bounded by the given surfaces. y = x2, z = 0, y + 2z = 4
> Express the integral ∭E f (x, y, z) dV as an iterated integral in six different ways, where E is the solid bounded by the given surfaces. y2 + z2 = 9, x = -2, x = 2
> Evaluate the iterated integral. ∫_0^2 ∫_0^(x^2) ∫_0^(y-2) (2x-y) dx dy dz
> Use cylindrical coordinates. Find the mass of a ball B given by x2 + y2 + z2 < a2 if the density at any point is proportional to its distance from the z-axis.
> Find the average value of the function f (x) = ∫_x^1cos(t^2) dt on the interval [0, 1].
> Use the Midpoint Rule for triple integrals (Exercise 24) to estimate the value of the integral. Divide B into eight sub-boxes of equal size. ∭B √x e xyz dV, where B = {(x, y, z) | 0 < x < 4, 0 < y < 1, 0 < z < 2} Exercise 24: (a). In the Midpoint Rule
> Use the Midpoint Rule for triple integrals (Exercise 24) to estimate the value of the integral. Divide B into eight sub-boxes of equal size. ∭B cos (xyz) dV, where B = {(x, y, z) | 0 < x < 1, 0 < y < 1, 0 < z < 1} Exercise 24: (a). In the Midpoint Rule
> Use cylindrical coordinates. Find the volume of the solid that lies between the paraboloid z = x2 + y2 and the sphere x2 + y2 + z2 = 2.
> Use cylindrical coordinates. Find the volume of the solid that is enclosed by the cone z = √(x^2 + y^2 ) and the sphere x2 + y2 + z2 = 2.
> Use cylindrical coordinates. Find the volume of the solid that lies within both the cylinder x2 + y2 = 1 and the sphere x2 + y2 + z2 = 4.
> Use cylindrical coordinates. Evaluate ∭E x2 dV, where E is the solid that lies within the cylinder x2 + y2 = 1, above the plane z = 0, and below the cone z2 = 4x2 + 4y2.
> Use cylindrical coordinates. Evaluate ∭E (x – y) dV, where E is the solid that lies between the cylinders x2 + y2 = 1 and x2 + y2 = 16, above the xy-plane, and below the plane z = y + 4.
> Plot the point whose cylindrical coordinates are given. Then find the rectangular coordinates of the point. (a). (√(2 ), 3π/4, 2) (b). (1, 1, 1)
> Use cylindrical coordinates. Evaluate ∭E (x + y + z) dV, where E is the solid in the first octant that lies under the paraboloid z = 4 - x2 - y2.
> Evaluate the integral where max {x2, y2} means the larger of the numbers x2 and y2. ,max(r".y") dy dx lo Jo
> Use cylindrical coordinates. Evaluate ∭E z dV, where E is enclosed by the paraboloid z = x2 + y2 and the plane z = 4.
> Use cylindrical coordinates. Evaluate ∭E √(x^2+ y^2 ) dV, where E is the region that lies inside the cylinder x2 + y2 = 16 and between the planes z = -5 and z = 4.
> Evaluate the triple integral. ∭E sin y dV, where E lies below the plane z = x and above the triangular region with vertices (0, 0, 0), (π, 0, 0), and (0, π, 0)
> Evaluate the triple integral. ∭E z/(x^2+y^2 ) dV, where E = {(x, y, z) | 1 < y < 4, y < z < 4, 0 < x < z j
> Evaluate the integral in Example 1, integrating first with respect to y, then z, and then x. Example 1: B = {(x, y, z) | 0 < x< 1, -1<y< 2, 0< z < 3}
> Evaluate the iterated integral. ∫_0^π ∫_0^1 ∫_0^(√(1-z^2 )) z sin x dy dz dx
> Sketch the solid whose volume is given by the iterated integral. ∫_0^2 ∫_0^(2-y) ∫_0^(4-y^2) dx dz dy
> Sketch the solid whose volume is given by the iterated integral. ∫_0^1 ∫_0^(1-x) ∫_0^(2-2z)〖dy dz dx〗
> (a). In the Midpoint Rule for triple integrals we use a triple Riemann sum to approximate a triple integral over a box B, where f (x, y, z) is evaluated at the center (x ̅i, y ̅j, z ̅k) of the box Bijk. Use the Midpoint Rule to estimate ∭B √(x^2 + y^2
> (a). Express the volume of the wedge in the first octant that is cut from the cylinder y2 1 z2 − 1 by the planes y = x and x = 1 as a triple integral. (b). Use either the Table of Integrals (on Reference Pages 6–10) or a computer algebra system to find t
> Evaluate lim┬(n→∞)n^(-2) ∑_(i=1)^n ∑_(j=1)^(n^2) 1/√(n^2+ni+j)
> Use a triple integral to find the volume of the given solid. The solid enclosed by the cylinder x2 + z2 = 4 and the planes y = -1 and y + z = 4
> Use a triple integral to find the volume of the given solid. The solid enclosed by the cylinder y = x2 and the planes z = 0 and y + z = 1
> Find, to four decimal places, the area of the part of the surface z = (1 + x2) / (1 + y2) that lies above the square |x | + |y | < 1. Illustrate by graphing this part of the surface.
> Evaluate the integral ∭E (xy + z2) dV, where E = {(x, y, z) | 0 < x < 2, 0 < y < 1, 0 < z < 36 using three different orders of integration.
> Find, to four decimal places, the area of the part of the surface z = 1 + x2y2 that lies above the disk x2 + y2 < 1.
> Evaluate the triple integral. ∭E z dV, where E is bounded by the cylinder y2 + z2 = 9 and the planes x = 0, y = 3x, and z = 0 in the first octant.
> Evaluate the triple integral. ∭E x dV, where E is bounded by the paraboloid x = 4y2 + 4z2 and the plane x = 4
> (a). Use the Midpoint Rule for double integrals with m = n = 2 to estimate the area of the surface z = xy + x2 + y2, 0 < x < 2, 0 < y < 2. (b). Use a computer algebra system to approximate the surface area in part (a) to four decimal places. Compare wit
> Find the area of the surface. The part of the surface z = xy that lies within the cylinder x2 + y2 = 1
> Find the area of the surface. The surface z =2/3 (x3/2 + y3/2), 0 < x < 1, 0 < y < 1
> If the ellipse x2/a2 + y2/b2 = 1 is to enclose the circle x2 + y2 = 2y, what values of a and b minimize the area of the ellipse?
> Find the area of the surface. The part of the hyperbolic paraboloid z = y2 - x2 that lies between the cylinders x2 + y2 = 1 and x2 + y2 = 4
> Find the area of the surface. The part of the cylinder x2 + z2 = 4 that lies above the square with vertices (0, 0), (1, 0), (0, 1), and (1, 1)
> Evaluate the integral by reversing the order of integration. ∫_0^2 ∫_(y/2)^1y cos (x3 – 1) dx dy
> Evaluate the integral by reversing the order of integration. ∫_0^1 ∫_(√x)^1√(y^3+1) dy dx
> Evaluate the integral by reversing the order of integration. ∫_0^1 ∫_(x^2)^1√y sin y dy dx
> Sketch the region of integration and change the order of integration. ∫_0^1 ∫_arctanx^(π/4) f (x,y) dy dx
> Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ. D is the triangular region with vertices (0, 0), (2, 1), (0, 3); ρ (x, y) = x + y
> Sketch the region of integration and change the order of integration. ∫_1^2 ∫_0^lnxf (x,y) dy dx
> Sketch the region of integration and change the order of integration. ∫_(-2)^2 ∫_0^(√(4-y^2 ) f (x,y) dx dy
> Sketch the region of integration and change the order of integration. ∫_0^(π/2) ∫_0^cosxf (x,y) dy dx
> Sketch the region of integration and change the order of integration. ∫_0^2 ∫_(x^2)^4 f (x,y) dy dx
> Sketch the region of integration and change the order of integration. ∫_0^1 ∫_0^y f (x,y) dx dy
> Calculate the iterated integral ∫(-3)^3 ∫_0^(π/2) (y+y^2 cos x ) dx dy
> Use a computer algebra system to find the exact volume of the solid. Enclosed by z = 1 - x2 - y2 and z = 0
> Use a computer algebra system to find the exact volume of the solid. Between the paraboloids z = 2x2 + y2 and z = 8 - x2 - 2y2 and inside the cylinder x2 + y2 = 1
> Use the result of Exercise 40 part (c) to evaluate the following integrals. Exercise 40 part (c): Deduce that (a). ∫_0^∞ x^2 e^(-x^2 ) dx (b). ∫_0^∞ √x e-x dx
> (a). We define the improper integral (over the entire plane R2) where Da is the disk with radius a and center the origin. Show that (b). An equivalent definition of the improper integral in part (a) is where Sa is the square with vertices (Â&p
> Find the area of the surface. The part of the surface 2y + 4z - x2 = 5 that lies above the triangle with vertices (0, 0), (2, 0), and (2, 4)
> Use polar coordinates to combine the sum into one double integral. Then evaluate the double integral. Sa l ty dy dx + ftvdy dx + 4-x2 xy dy dx ху.
