(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 the exact value of the triple integral in part (a).
Table from Pages 6–10:
Year …………………………………………………… Population (millions)
1900 …………………………………………………………………………… 1650
1910 …………………………………………………………………………… 1750
1920 …………………………………………………………………………… 1860
1930 …………………………………………………………………………… 2070
1940 …………………………………………………………………………… 2300
1950 …………………………………………………………………………… 2560
1960 …………………………………………………………………………… 3040
1970 …………………………………………………………………………… 3710
1980 …………………………………………………………………………… 4450
1990 …………………………………………………………………………… 5280
2000 …………………………………………………………………………… 6080
2010 …………………………………………………………………………… 6870
> 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 dθ 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 dθ 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)
> 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
> 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
> If you attempt to use Formula 2 to find the area of the top half of the sphere x2 + y2 + z2 = a2, you have a slight problem because the double integral is improper. In fact, the integrand has an infinite discontinuity at every point of the boundary circl
> Show that the area of the part of the plane z = ax + by + c that projects onto a region D in the xy-plane with area A (D) is √(a^2 + b^2 + 1) A (D).
> Find the area of the surface. The part of the plane 6x + 4y + 2z = 1 that lies inside the cylinder x2 + y2 = 25
> Find the exact area of the surface z = 1 + x + y + x2 -2 < x < 1 -1 < y < 1 Illustrate by graphing the surface.
> Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1 < x < 4, 0 < y < 1.
> (a). Use the Midpoint Rule for double integrals (see Section 15.1) with four squares to estimate the surface area of the portion of the paraboloid z = x2 + y2 that lies above the square [0, 1] × [0, 1]. (b). Use a computer algebra system to approximate t
> Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral. The part of the surface z = cos (x2 + y2) that lies inside the cylinder x2 + y2 = 1