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mathematics
precalculus

Questions and Answers of Precalculus

Let E be the solid in the first octant bounded by the cylinder x2 + y2 = 1 and the planes y = z, x = 0, and z = 0 with the density function p(x, y, z) = 1 + x + y + z. Use a computer algebra system
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; p(x, y, z) = √x2 + y2
Assume that the solid has constant density k.Find the moment of inertia about the z-axis of the solid cone √x2 + y2 < z < h.
Assume that the solid has constant density k.Find the moment of inertia about the z-axis of the solid cylinder x2 + y2 < a2, 0 < z < h.
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
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 p.E is the tetrahedron bounded by the planes x = 0, y = 0, z = 0, x + y + z = 1; p(x, y, z) = y
Find the mass and center of mass of the solid E with the given density function p.E is the cube given by 0 < x < a, 0 < y < a, 0 < z < a; p(x, y, z) = x2 + y2 + z2
Find the mass and center of mass of the solid E with the given density function p.E is bounded by the parabolic cylinder z = 1 - y2 and the planes x + z = 1, x = 0, and z = 0; p(x, y, z) = 4
Find the mass and center of mass of the solid E with the given density function p.E lies above the xy-plane and below the paraboloid z = 1 - x2 - y2 ; p(x, y, z) = 3
Evaluate the triple integral using only geometric interpretation and symmetry.where B is the unit ball x2 + y2 + z2 < 1 SIl: (z³ + sin y + 3) dV,
Evaluate the triple integral using only geometric interpretation and symmetry.where C is the cylindrical region x2 + y2 < 4, -2 < z < 2 II. (4 + 5x²yz?) dV,
Write five other iterated integrals that are equal to the given iterated integral. f(x, y, z) dx dz dy Jy
Write five other iterated integrals that are equal to the given iterated integral. f(x, y, z) dz dx dy 0 Jy
The figure shows the region of integration for the integral
The figure shows the region of integration for the integral
Express the integral as an iteratedintegral in six different ways, where E is the solid bounded by the given surfaces.x = 2, y = 2, z = 0, x + y - 2z = 2 ГУУE F(x, y, 2) dV
Express the integral as an iteratedintegral in six different ways, where E is the solid bounded by the given surfaces.y = x2, z − 0, y + 2z = 4 ГУУE F(x, y, 2) dV
Express the integral as an iteratedintegral in six different ways, where E is the solid bounded by the given surfaces.y2 + z2 = 9, x = -2, x = 2 ГУУE F(x, y, 2) dV
Express the integral as an iteratedintegral in six different ways, where E is the solid bounded by the given surfaces.y = 4 - x2 - 4z2, y = 0 ГУУE F(x, y, 2) dV
Sketch the solid whose volume is given by the iterated integral. (2 (2-y (4-y² dx dz dy
Sketch the solid whose volume is given by the iterated integral. C1-x C2-2z dy dz dx 0 Jo
Use the Midpoint Rule for triple integrals (Exercise 24) to estimate the value of the integral. Divide B into eight subboxes of equal size. Sle Vxe*y= dV, where B = {(x, y, z) | 0 < x < 4, 0 < y < 1,
Use the Midpoint Rule for triple integrals (Exercise 24) to estimate the value of the integral. Divide B into eight subboxes of equal size. SII, cos(xyz) dV, where B = {(x, y, z) | 0 < x < 1, 0 < y <
(a) Express the volume of the wedge in the first octant that is cut from the cylinder y2 + z2 = 1 by the planes y = x and x = 1 as a triple integral. (b) Use either the Table of Integrals (on
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
Use a triple integral to find the volume of the given solid.The solid enclosed by the paraboloids y = x2 + z2 and y = 8 - x2 - z2
Use a triple integral to find the volume of the given solid.The tetrahedron enclosed by the coordinate planes and the plane 2x + y + z = 4
Evaluate the triple integral.where E is bounded by the paraboloid x = 4y2 + 4z2 and the plane x = 4 ГУЕх dV, JE
Evaluate the triple integral.where E is bounded by the cylinder y2 + z2 = 9 and the planes x = 0, y = 3x, and z = 0 in the first octant SIe z dV, JE
Evaluate the triple integral.where T is the solid tetrahedron with vertices (0, 0, 0), (1, 0, 1), (0, 1, 1), and (0, 0, 1) SII, xz dV,
Evaluate the triple integral.where E is enclosed by the surfaces z = x2 - 1, z = 1 - x2, y = 0, and y = 2 SS: (x – y) dV,
Evaluate the triple integral.where T is the solid tetrahedron with vertices (0, 0, 0), (2, 0, 0), (0, 2, 0), and (0, 0, 2) dV
Evaluate the triple integral.where E lies under the plane z = 1 + x + y and above the region in the xy-plane bounded by the curves y = √x , y = 0, and x = 1 ГУ 6ху dV,
Evaluate the triple integral.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) Г 6ху dV, JE
Evaluate the triple integral.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) SSSE sin y dV,
Evaluate the triple integral. SSLE e:/dV, where xy} E = {(x, y, z) | 0 < y< 1, y < x < 1, 0 < z <
Evaluate the triple integral. SILE y dV, where E = {(x, y, z) | 0 < x < 3,0 < y < x, x – y< z< x + y}
Evaluate the iterated integral. ГЕС 2z ln x хе dy dx dz
Evaluate the iterated integral. CCCxye dz dy dx Ci(2-x2-y2 Jo Jo Jo
Evaluate the iterated integral. V1-z2 z sin x dy dz dx Jo Jo
Evaluate the iterated integral. 1-z2 dx dz dy 0 Jo y + 1
Evaluate the iterated integral. *2y (x+y бху dz dx dy Jo Jy
Evaluate the iterated integral. y-z (2х — у) dx dy dz Vo Jo
Evaluate the integralE = 5(x, y, z) | 0 < x < 2, 0 < y < 1, 0 < z < 36 using three different orders of integration. ЛУ. (ху + 2?) dV, E
Evaluate the integral in Example 1, integrating first with respect to y, then z, and then x.
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
The figure shows the surface created when the cylinder y2 + z2 = 1 intersects the cylinder x2 + z2 = 1. Find the area of this surface.
Find the area of the finite part of the paraboloid y = x2 + z2 cut off by the plane y = 25.
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 √a2 + b2 + 1 A(D).
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.
Find, to four decimal places, the area of the part of the surface z = 1 + x2y2 that lies above the disk x2 + y2 < 1.
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 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
(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] x [0,
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
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 = 1/(1 +
Find the area of the surface.The part of the sphere x2 + y2 + z2 = 4z that lies inside the paraboloid z = x2 + y2
Find the area of the surface.The part of the sphere x2 + y2 + z2 = a2 that lies within the cylinder x2 + y2 = ax and above the xy-plane
Find the area of the surface.The part of the sphere x2 + y2 + z2 = 4 that lies above the plane z = 1
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
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)
Find the area of the surface.The part of the paraboloid z = 1 - x2 - y2 that lies above the plane z = -2
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)
Find the area of the surface.The part of the plane 3x + 2y + z = 6 that lies in the first octant
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 area of the surface.The part of the plane 5x + 3y - z + 6 = 0 that lies above the rectangle f1, 4] x [2, 6]
When studying the spread of an epidemic, we assume that the probability that an infected individual will spread the disease to an uninfected individual is a function of the distance between them.
Xavier and Yolanda both have classes that end at noon and they agree to meet every day after class. They arrive at the coffee shop independently. Xavier’s arrival time is X and Yolanda’s arrival
Suppose that X and Y are independent random variables,where X is normally distributed with mean 45 and standarddeviation 0.5 and Y is normally distributed with mean 20and standard deviation 0.1.(a)
(a) A lamp has two bulbs, each of a type with average lifetime 1000 hours. Assuming that we can model the probability of failure of a bulb by an exponential density function with mean µ = 1000, find
The joint density function for a pair of random variables X and Y is(a) Find the value of the constant C.(b) Find P(X < 1, Y < 1).(c) Find P(X + Y < 1). Cx(1 + y) if 0
Consider a square fan blade with sides of length 2 and the lower left corner placed at the origin. If the density of the blade is (x, y) = 1 + 0.1x, is it more difficult to rotate the blade about
Find the moments of inertia Ix, Iy, I0 for the lamina of Exercise 15.
Find the moments of inertia Ix, Iy, I0 for the lamina of Exercise 6.
Find the moments of inertia Ix, Iy, I0 for the lamina of Exercise 3.
A lamina occupies the region inside the circle x2 + y2 = 2y but outside the circle x2 + y2 = 1. Find the center of mass if the density at any point is inversely proportional to its distance from the
Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length a if the density at any point is proportional to the square of the distance from the vertex
Find the center of mass of the lamina in Exercise 13 if the density at any point is inversely proportional to its distance from the origin.
The boundary of a lamina consists of the semicircles y = √1 - x2 and y = √4 - x2 together with the portions of the x-axis that join them. Find the center of mass of the lamina if the density at
Find the center of mass of the lamina in Exercise 11 if the density at any point is proportional to the square of its distance from the origin.
A lamina occupies the part of the disk x2 + y2 < 1 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.
Find the mass and center of mass of the lamina that occupies the region D and has the given density function p.D = h(x, y)| 0 < x < a, 0 < y < b}; (x, y) = 1 + x2 + y2
Find the mass and center of mass of the lamina that occupies the region D and has the given density function p.D = h(x, y)| 1 < x < 3, 1 < y < 4}; (x, y) = ky2
Electric charge is distributed over the disk x2 + y2 < 1 so that the charge density at (x, y) is (x, y) = √x2 + y2 (measured in coulombs per square meter). Find the total charge on the
Electric charge is distributed over the rectangle 0 < x < 5, 2 < y < 5 so that the charge density at (x, y) is (sx, y) = 2x + 4y (measured in coulombs per square meter). Find the total
Use the result of Exercise 40 part (c) to evaluate the following integrals.(a)(b) x²e** dx Vxe*dx хе
(a) We define the improper integral (over the entire plane R2d)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)
Use polar coordinates to combine the suminto one double integral. Then evaluate the double integral. +*[»vdv dx + f /4-х2 Se S ty dy dx + ty dy dx + ху ху dy dx J1//2 J1-x2
Let D be the disk with center the origin and radius a. What is the average distance from points in D to the origin?
Find the average value of the functionon the annular region a2 < x2 + y2 < b2, where 0 < a < b. F(x, y) = 1//x² + y²
An agricultural sprinkler distributes water in a circular pattern of radius 100 ft. It supplies water to a depth of e-r feet per hour at a distance of r feet from the sprinkler.(a) If 0 < R <
A swimming pool is circular with a 40-ft diameter. The depth is constant along east-west lines and increases linearly from 2 ft at the south end to 7 ft at the north end. Find the volume of water in
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.where D is the portion of the disk x2 +
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.where D is the disk with center the
Evaluate the iterated integral by converting to polar coordinates. V2x-x2 Vx2 + y2 dy dx 0.

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