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 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 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.
> Calculate the iterated integral ∫_0^1 ∫_1^2(x+e^(-y) dx dy
> (a). Newton’s method for approximating a root of an equation f (x) = 0 (see Section 4.8) can be adapted to approximating a solution of a system of equations f (x, y) = 0 and g (x, y) = 0. The surfaces z = f (x, y) and z = t (x, y) inter
> 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) %3 ху; 4х? + y? %3D 8 = 8
> 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 e^(x^2+y^2 )^2 dA, where D is the disk with center the origin and radius 1
> Evaluate the iterated integral by converting to polar coordinates. ∫_0^2 ∫_0^(√(2x-x^2 ) √(x^2+y^2 ) dy dx
> Evaluate the iterated integral by converting to polar coordinates. ∫_0^(1/2) ∫_(√3 y)^(√(1-y^2 ) xy^2 dx dy
> Evaluate the iterated integral by converting to polar coordinates. ∫_0^a ∫_(-√(a^2-y^2 ))^(√(a^2-y^2 ) (2x + y) dx dy
> Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ. D = {(x, y) | 1 < x < 3, 1 < y < 4}; ρ (x, y) = ky2
> Evaluate the iterated integral by converting to polar coordinates. ∫_0^2 ∫_0^(√(4-x^2 ) e^(-x^2-y^2 ) dy dx
> Find the volume of the given solid. Bounded by the planes z = x, y = x, x + y = 2, and z = 0
> Use polar coordinates to find the volume of the given solid. Inside both the cylinder x2 + y2 = 4 and the ellipsoid 4x2 + 4y2 + z2 = 64
> Use a computer algebra system to find the mass, center of mass, and moments of inertia of the lamina that occupies the region D and has the given density function. D is enclosed by the right loop of the four-leaved rose r = cos 2θ; θ (x, y) = x2 + y2
> (a). Find the maximum value of f (x1, x2, . . . , xn) = √(n&x_1 x_2…x_n ) given that x1, x2, . . ., xn are positive numbers and x1 + x2 + ∙ ∙ ∙ + xn = c, wher
