An important problem in thermodynamics is to find the work done by an ideal Carnot engine. A cycle consists of alternating expansion and compression of gas in a piston. The work done by the engine is equal to the area of the region R enclosed by two isothermal curves xy = a, xy = b and two adiabatic curves xy1.4 = c, xy1.4 = d, where 0 < a < b and 0 < c < d. Compute the work done by determining the area of R.
> Sketch the solid whose volume is given by the iterated integral. ∫_0^1 ∫_0^1〖(2 -x^2 - y^2)〗dx dy
> Graph the solid bounded by the plane x + y + z = 1 and the paraboloid z = 4 - x2 - y2 and find its exact volume. (Use your CAS to do the graphing, to find the equations of the boundary curves of the region of integration, and to evaluate the double integ
> Use geometry or symmetry, or both, to evaluate the double integral. ∬D (ax3 + by3 + √(a^2 - x^2 ) d dA, D = [-a, a] × [-b, b]
> Use geometry or symmetry, or both, to evaluate the double integral. ∬D (2 + x2y3 - y2 sin x) dA, D = {(x, y) | |x | + |y | < 1j
> Use geometry or symmetry, or both, to evaluate the double integral. (2x + 3y) dA, D is the rectangle 0 < x < a, 0 < y < b
> Use geometry or symmetry, or both, to evaluate the double integral. ∬D √(R^2-x^2-y^2 ) dA, D is the disk with center the origin and radius R
> Use geometry or symmetry, or both, to evaluate the double integral. ∬D (x + 2) dA, D = {(x, y) | 0 < y < √(9 - x^2 )
> Find the average value of the function f (x, y) = 1/√(x^2 + y^2 ) on the annular region a2 < x2 + y2 < b2, where 0 < a < b.
> Prove Property 11. Property 11: If m < f(x, y) < M for all (x, y) in D, then mA(D) f(x, y) dA < MA(D)
> (a). If your computer algebra system plots implicitly defined curves, use it to estimate the minimum and maximum values of f (x, y) = x3 + y3 + 3xy subject to the constraint (x – 3)2 + (y – 3)2 = 9 by graphical methods. (b). Solve the problem in part (a)
> An agricultural sprinkler distributes water in a circular pattern of radius 100 ft. It supplies water to a depth of e2r feet per hour at a distance of r feet from the sprinkler. (a). If 0 < R < 100, what is the total amount of water supplied per hour to
> Find the averge value of f over the region D. f (x, y) = xy, D is the triangle with vertices (0, 0), (1, 0), and (1, 3)
> Use Property 11 to estimate the value of the integral. ∬T sin4(x + y) dA, T is the triangle enclosed by the lines y = 0, y = 2x, and x = 1 Property 11: If m < f(x, y) < M for all (x, y) in D, then mA(D) f(x, y) dA < MA(D)
> Use Property 11 to estimate the value of the integral. ∬S √(4- x^2 y^2 ) dA, S = {(x, y) | x2 + y2 0} Property 11: If m < f(x, y) < M for all (x, y) in D, then mA(D) f(x, y) dA < MA(D)
> Find the approximate volume of the solid in the first octant that is bounded by the planes y = x, z = 0, and z = x and the cylinder y = cos x. (Use a graphing device to estimate the points of intersection.)
> Express D as a union of regions of type I or type II and evaluate the integral. ∬D x2 dA yA 1 (1, 1) D -1 1 -1
> (a). Find the region E for which the triple integral is a maximum. (b). Use a computer algebra system to calculate the exact maximum value of the triple integral in part (a). (1 – x? – 2y? – 3z²) dV – 32') dV
> Find the moments of inertia Ix, Iy, I0 for the lamina of Exercise 15. Exercise 15: 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
> Consider the problem of minimizing the function f (x, y) = x on the curve y2 + x4 - x3 = 0 (a piriform). (a). Try using Lagrange multipliers to solve the problem. (b). Show that the minimum value is f (0, 0) = 0 but the Lagrange condition ∆f (0, 0) = ∆t
> (a). In what way are the theorems of Fubini and Clairaut similar? (b). If f (x, y) is continuous on [a, b] × [c, d] and glx, v) = [" [ f(s, o) dt ds for a <x< b, c<y<d, show that gry = gyx = f(x, y).
> The joint density function for random variables X, Y, and Z is f (x, y, z) = Cxyz if 0 < x < 2, 0 < y < 2, 0 < z < 2, and f (x, y, z) = 0 otherwise. (a). Find the value of the constant C. (b). Find P (X < 1, Y < 1, Z < 1). (c). Find P (X + Y 1 Z < 1).
> Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why it’s easier. ∬D y dA, D is bounded by y = x - 2, x = y2
> (a). Use cylindrical coordinates to show that the volume of the solid bounded above by the sphere r2 + z2 = a2 and below by the cone z = r cot φ0 (or φ = φ0), where 0 (b). Deduce that the volume of the spherical wedge g
> (a). A cylindrical drill with radius r1 is used to bore a hole through the center of a sphere of radius r2. Find the volume of the ring-shaped solid that remains. (b). Express the volume in part (a) in terms of the height h of the ring. Notice that the v
> Graph the solid that lies between the surface z = 2xy/ (x2 + 1) and the plane z = x + 2y and is bounded by the planes x = 0, x = 2, y = 0, and y = 4. Then find its volume.
> Use a graphing device to draw a silo consisting of a cylinder with radius 3 and height 10 surmounted by a hemisphere.
> Use a computer algebra system to find the exact volume of the solid. Enclosed by z = x2 + y2 and z = 2y
> Find the volume of the solid enclosed by the paraboloid z = 2 + x2 + (y – 2)2 and the planes z = 1, x = 1, x = 21, y = 0, and y = 4.
> Sketch the solid whose volume is given by the iterated integral. ∫_0^1 ∫_0^1(4 - x - 2y) dx dy
> Evaluate the integral by changing to spherical coordinates ∫_0^1 ∫_0^(√(1-x^2 )) ∫_(√(x^2+y^2 ))^(√(2-x^2-y^2 )) xy dz dy dx
> Find the volume of the solid by subtracting two volumes. The solid enclosed by the parabolic cylinders y = 1 - x2, y = x2 - 1 and the planes x + y + z = 2, 2x + 2y – z+1 10 = 0
> Sketch the solid whose volume is given by the iterated integral. ∫_0^1 ∫_0^(1-x)(1 - x – y) dy dx
> Use cylindrical or spherical coordinates, whichever seems more appropriate. A solid right circular cone with constant density has base radius a and height h. (a). Find the moment of inertia of the cone about its axis. (b). Find the moment of inertia of t
> Use cylindrical or spherical coordinates, whichever seems more appropriate. A solid cylinder with constant density has base radius a and height h. (a). Find the moment of inertia of the cylinder about its axis. (b). Find the moment of inertia of the cyli
> Use cylindrical or spherical coordinates, whichever seems more appropriate. Find the volume of the smaller wedge cut from a sphere of radius a by two planes that intersect along a diameter at an angle of π/6.
> 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 the pool.
> The figure shows the region of integration for the integral ∫_0^1 ∫_(√x)^(1-x^2) ∫_0^(1-x) f (x,y,z) dy dz dx Rewrite this integral as an equivalent iterated integral in the five
> Use spherical coordinates. (a). Find the centroid of a solid homogeneous hemisphere of radius a. (b). Find the moment of inertia of the solid in part (a) about a diameter of its base.
> Find the extreme values of f on the region described by the inequality. f(x, y) = e-", x² + 4y² < 1
> 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 time is Y, where X and
> When studying the formation of mountain ranges, geologists estimate the amount of work required to lift a mountain from sea level. Consider a mountain that is essentially in the shape of a right circular cone. Suppose that the weight density of the mater
> Evaluate the integral by changing to cylindrical coordinates. ∫_(-3)^3 ∫_0^(√(9-x^2 ) ∫_0^(9-x^2-y^2) √(x^2+y^2 ) dz dy dx
> (a). Verify that f (x, y) = {_0^4xy if 0 < x < 1, 0 < y < 1 otherwise is a joint density function. (b). If X and Y are random variables whose joint density function is the function f in part (a), find (i). P (X > 1 2 ) (ii). P (X > 1 2 , Y < 1 2 ) (c
> Evaluate the integral by making an appropriate change of variables. ∬R ex+y dA, where R is given by the inequality |x | + |y | < 1
> Use cylindrical coordinates. (a). Find the volume of the solid that the cylinder r = a cos cuts out of the sphere of radius a centered at the origin. (b). Illustrate the solid of part (a) by graphing the sphere and the cylinder on the same screen.
> Use cylindrical coordinates. (a). Find the volume of the region E that lies between the paraboloid z = 24 - x2 - y2 and the cone z = 2 √(x^2 + y^2 ). (b). Find the centroid of E (the center of mass in the case where the density is constant).
> Evaluate the integral by making an appropriate change of variables. ∬R (x + y)e^(x^2-y^2 ) dA, where R is the rectangle enclosed by the lines x - y = 0, x - y = 2, x + y = 0, and x + y = 3
> Evaluate the integral by making an appropriate change of variables. ∬R (x-2y)/(3x-y) dA, where R is the parallelogram enclosed by the lines x - 2y = 0, x - 2y = 4, 3x - y = 1, and 3x - y = 8
> A rectangle with length L and width W is cut into four smaller rectangles by two lines parallel to the sides. Find the maximum and minimum values of the sum of the squares of the areas of the smaller rectangles.
> (a). Evaluate ∭E dV, where E is the solid enclosed by the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1. Use the transformation x = au, y = bv, z = cw. (b). The earth is not a perfect sphere; rotation has resulted in flattening at the poles. So, the shape can be a
> 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
> Use a double integral to find the area of the region. The region inside the circle (x – 1)2 + y2 = 1 and outside the circle x2 + y2 = 1
> Use the given transformation to evaluate the integral. ∬R (4x + 8y) dA, where R is the parallelogram with vertices (-1, 3), (1, -3), (3, -1), and (1, 5); x = 1/4 (u + v), y = 1/4 (v - 3u)
> Use the given transformation to evaluate the integral. ∬R (x - 3y) dA, where R is the triangular region with vertices (0, 0), (2, 1), and (1, 2); x = 2u + v, y = u + 2v
> A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes. R is bounded by the hyperbolas y = 1/x, y = 4/x and the lines y =
> A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes. R lies between the circles x2 + y2 = 1 and x2 + y2 = 2 in the firs
> Find the extreme values of f on the region described by the inequality. f(x, y) = x² + y² + 4x – 4y, x? + y? < 9
> A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes. R is the parallelogram with vertices (0, 0), (4, 3), (2, 4), (-2,
> A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes. R is bounded by y = 2x - 1, y = 2x + 1, y = 1 - x, y = 3 - x
> Write the equation in spherical coordinates. (a). z = x2 + y2 (b). z = x2 - y2
> Find the image of the set S under the given transformation. S is the triangular region with vertices (0, 0), (1, 1), (0, 1); x = u2, y = v
> Find the image of the set S under the given transformation. S is the square bounded by the lines u = 0, u = 1, v = 0, v = 1; x = v, y = u (1 + v2)
> Find the image of the set S under the given transformation. S = {(u, v) | 0 < u < 3, 0 < v < 2}; x = 2u + 3v, y = u - v
> Describe in words the surface whose equation is given. ρ2 - 3ρ + 2 = 0
> Describe in words the surface whose equation is given. φ = π/3
> Change from rectangular to cylindrical coordinates. (a). (-√(2 ), √(2 ), 1) (b). (2, 2, 2)
> Change from rectangular to cylindrical coordinates. (a). (-1, 1, 1) (b). (-2, 2 3 , 3)
> Find the extreme values of f subject to both constraints. f(x, y, z) = x² + y² + z*; x – y = 1, y² – z² = 1
> (a). Use a graphing calculator or computer to graph the circle x2 + y2 = 1. On the same screen, graph several curves of the form x2 + y = c until you find two that just touch the circle. What is the significance of the values of c for these two curves? (
> Plot the point whose cylindrical coordinates are given. Then find the rectangular coordinates of the point. (a). (4, π/3, -2) (b). (2, - π/2, 1)
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. 2. There exists a function f with continuous second-order partial derivatives such that fx (x,
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 5. If f is continuous on [0, 1], then 7. If D is the disk given by x2 + y2 8. The integra
> Sketch the graph of the function. f (x, y) = x2 + (y – 2)2
> Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s). f(x, y - 1 y y
> Use a graphing calculator or computer (or Newton’s method or a computer algebra system) to find the critical points of f (x, y) = 12 + 10y - 2x2 - 8xy - y4 correct to three decimal places. Then classify the critical points and find the highest point on t
> Use a graph or level curves or both to estimate the local maximum and minimum values and saddle points of f (x, y) = x3 - 3x + y4 - 2y2. Then use calculus to find these values precisely.
> Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. f (x, y) = x3 - 6xy + 8y3
> Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. f (x, y) = x2 - xy + y2 +
> A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the figure. If the pentagon has fixed perimeter P, find the lengths of the sides of the pentagon that maximize the area of the pentagon.
> Shoe that ∫_0^∞(arctan πx-arctanx)/x dx=π/2 lnπ by first expressing the integral as an iterated integral.
> A package in the shape of a rectangular box can be mailed by the US Postal Service if the sum of its length and girth (the perimeter of a cross-section perpendicular to the length) is at most 108 in. Find the dimensions of the package with largest volume
> Find the points on the surface xy2z3 = 2 that are closest to the origin.
> Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s). f (x, y, z) = x2 + 2y2 + 3z2; x + y + z = 1, x - y + 2z = 2
> Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s). f (x, y, z) = xyz; x2 + y2 + z2 = 3
> (a). Evaluate ∬D 1/((x^2+y^2 )^(n/2) ) dA, where n is an integer and D is the region bounded by the circles with center the origin and radii r and R, 0 < r < R. (b). For what values of n does the integral in part (a) have a limit as r → +? (c). Find ∭E 1
> Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s). f (x, y) = x2y; x2 + y2 = 1
> The Mean Value Theorem for double integrals says that if f is a continuous function on a plane region D that is of type I or II, then there exists a point sx0, y0 d in D such that Use the Extreme Value Theorem (14.7.8) and Property 15.2.11 of integrals
> Use the change of variables formula and an appropriate transformation to evaluate ∬R xy dA, where R is the square with vertices (0, 0), (1, 1), (2, 0), and (1, -1).
> Find the absolute maximum and minimum values of f on the set D. f (x, y) = e-x2-y2 (x2 + 2y2); D is the disk x2 + y2 < 4
> Find the absolute maximum and minimum values of f on the set D. f (x, y) = 4xy2 - x2y2 - xy3; D is the closed triangular region in the xy-plane with vertices (0, 0), (0, 6), and (6, 0)
> Suppose f is a differentiable function of one variable. Show that all tangent planes to the surface z = x f (y/x) intersect in a common point.
> Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. f (x, y) = (x2 + y) ey/2
> Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. f (x, y) = 3xy - x2y - xy2
> A lamp has three bulbs, each of a type with average lifetime 800 hours. If we model the probability of failure of a bulb by an exponential density function with mean 800, find the probability that all three bulbs fail within a total of 1000 hours.
> The joint density function for random variables X and Y is (a). Find the value of the constant C. (b). Find P (X 1). (c). Find P (X + Y S(x + y) if 0 <I< 3, 0< y< 2 f(x, y) = otherwise
> Find parametric equations of the tangent line at the point (-2, 2, 4) to the curve of intersection of the surface z = 2x2 - y2 and the plane z = 4.
> The contour map shows wind speed in knots during Hurricane Andrew on August 24, 1992. Use it to estimate the value of the directional derivative of the wind speed at Homestead, Florida, in the direction of the eye of the hurricane. 70 60 70 5565 U 8
> Use spherical coordinates to evaluate ∫_(-2)^2 ∫_0^(√(4-y^2 ) ∫_(-√(4-x^2-y^2 ) ^(√(4-x^2-y^2 ) y^2 √(x^2+y^2+z^2 ) dz dx dy
> Use polar coordinates to evaluate ∫_0^3 ∫ (-√(9-x^2 ) ^(√(9-x^2 ) (x^2+xy^2) dy dx
