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. Use a triple integral and trigonometric substitution to find the volume of a sphere with radius r. 3. Use a quadruple integral to find the (4-dimensional) volume enclosed by the hypersphere x2 1 y2 + z2 + w2 = r2 in R4. (Use only trigonometric substitution and the reduction formulas for ∫ sin^n x dx or ∫ cos^n x dx.) 4. Use an n-tuple integral to find the volume enclosed by a hypersphere of radius r in n dimensional space Rn. [Hint: The formulas are different for n even and n odd.]
> If D is the region bounded by the curves y = 1/2 x2 and y = ex, find the approximate value of the integral ∬D y2 dA. (Use a graphing device to estimate the points of intersection of the curves.)
> Find the directional derivative of f at the given point in the indicated direction. f (x, y) = x2e-y, (-2, 0), in the direction toward the point (2, -3)
> Find the area of the part of the cone z2 = a2(x2 + y2) between the planes z = 1 and z = 2.
> Find the gradient of the function f (x, y, z) = x2eyz2.
> Calculate the value of the multiple integral. ∭H z3√(x^2+y^2+ z^2 ) dV, where H is the solid hemisphere that lies above the xy-plane and has center the origin and radius 1
> Calculate the value of the multiple integral. ∭E y2z2 dV, where E is bounded by the paraboloid x = 1 - y2 - z2 and the plane x = 0
> Calculate the value of the multiple integral. ∭T xy dV, where T is the solid tetrahedron with vertices (0, 0, 0), (1/3 , 0, 0), (0, 1, 0), and (0, 0, 1)
> Calculate the value of the multiple integral. ∬E xy dV, where E = {(x, y, z) | 0 < x < 3, 0 < y < x, 0 < z < x + y}
> Calculate the value of the multiple integral. ∭T x dA, where D is the region in the first quadrant that lies between the circles x2 1 y2 − 1 and x2 + y2 = 2
> Calculate the value of the multiple integral. ∭T (x2 +1 y2/3 dA, where D is the region in the first quadrant bounded by the lines y = 0 and y = √3 x and the circle x2 + y2 = 9
> 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) = xy²z; x² + y? + z? = 4
> Calculate the value of the multiple integral. ∬D y dA, where D is the region in the first quadrant that lies above the hyperbola xy = 1 and the line y = x and below the line y = 2
> Calculate the value of the multiple integral. ∬D y dA, where D is the region in the first quadrant bounded by the parabolas x = y2 and x = 8 - y2
> Describe the solid whose volume is given by the integral and evaluate the integra. (7/2 (T/2 S S" Sp° sino dp dộ dô
> Find the first partial derivatives. G (x, y, z) = exz sin (y/z)
> Find the first partial derivatives. F (α, β) = α2 ln (α2 + β2)
> Find the first partial derivatives. u + 20 u² + v? g(u, v)
> Find the first partial derivatives. f (x, y) = (5y3 + 2x2y)8
> Calculate the iterated integral. ∫_1^2 ∫_0^2 (y+2xe^y) dx dy
> Use the Midpoint Rule to estimate the integral in Exercise 1. Exercise 1: A contour map is shown for a function f on the square R = [0, 3] × [0, 3]. Use a Riemann sum with nine terms to estimate the value of ∬R f (x, y) dA
> Find and sketch the domain of the function. f (x, y) = ln (x + y + 1)
> 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) = e""; 2x² + y² + z? = 24
> Sketch several level curves of the function. f (x, y) = ex + y
> Sketch several level curves of the function. f (x, y) = √(4x^2 + y^2 )
> Find the direction in which f (x, y, z) = zexy increases most rapidly at the point (0, 1, 2). What is the maximum rate of increase?
> Find the maximum rate of change of f (x, y) = x2y + √("y" ) at the point (2, 1). In which direction does it occur?
> Find the directional derivative of f at the given point in the indicated direction. f (x, y, z) = x2y + x √(1 + z), (1, 2, 3), in the direction of v = 2i + j - 2k
> If cos (xyz) = 1 + x2y2 + z2, find ∂z/∂x and ∂z/∂x.
> If z = f (u, v), where u = xy, v = y/x, and f has continuous second partial derivatives, show that az + 2v dv -4uv ax? ởy? au dv
> The length x of a side of a triangle is increasing at a rate of 3 in/s, the length y of another side is decreasing at a rate of 2 in/s, and the contained angle θ is increasing at a rate of 0.05 radian/s. How fast is the area of the triangle changing when
> (a). Maximize ∑_(i -1)^n xi yi subject to the constraints (b). Put for any numbers a1, . . . , an, b1, . . . , bn. This inequality is known as the Cauchy-Schwarz Inequality. EL, x} = 1 and E y? = 1. bi and yi= di X; = /E b? to
> Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 49 14.7 Exercise 49: Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex
> 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) = 2x + 2y + z; x² + y? + z² = 9
> Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 46 14.7 Exercise 46: Find three positive numbers whose sum is 12 and the sum of whose squares is as small as possible.
> Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 45 14.7 Exercise 45: Find three positive numbers whose sum is 100 and whose product is a maximum.
> Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 41 Exercise 41: Find the shortest distance from the point (2, 0, -3) to the plane x + y + z = 1.
> Use Lagrange multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral. Hint: Use Heron’s formula for the area: where s = p/2 and x, y, z are the lengths of the sides. A = Vs(s – x)(s – y
> Referring to Exercise 27, we now suppose that the production is fixed at bLaK1-a = Q, where Q is a constant. What values of L and K minimize the cost function C (L, K) = mL + nK? Exercise 27: The total production P of a certain product depends on the a
> 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
> 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 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.
> 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)
