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mathematics
calculus 10th edition

Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions

Sketch the region R whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. 32 dx dy
Let the plane region R be a unit circle and let the maximum value of ƒ on R be 6. Is the greatest possiblevalue of ∫R ∫ ƒ(x, y) dy dx equal to 6? Why or why not? Ifnot, what is the greatest possible value?
Evaluate the iterated integral. TT/2 [I (4-z 0 0 z dr dz do
Let R be a region in the xy-plane whose area is B. When ƒ(x, y) = k for every point (x, y) in R, what is thevalue of ∫R∫ ƒ(x, y) dA? Explain.
Set up a triple integral that gives the moment of inertia about the z-axis of the solid region Q of density ρ. Q = {(x, y, z): x² + y² ≤ 1,0 ≤ z ≤ 4 - x² - y²} p = kx²
Horizontal cross sections of a piece of ice that broke from a glacier are in the shape of a quarter of a circle with a radius of approximately 50 feet. The base is divided into 20 subregions, as shown in the figure. At the center of each subregion, the height of the ice is measured, yielding the
Use a computer algebra system to evaluate the iterated integral. 2y 0 Jy sin(x + y) dx dy
Determine the region R in the xy-plane that maximizes the value of So. JR. (9-x² - y²) dA.
Evaluate the iterated integral. *π/2 (π/2 [ "p² dpc p² dp de do P
Sketch the region R whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. 2 4-y² -2. -2J0 dx dy
Use a computer algebra system to approximate the iterated integral. #/4 (4 S Jo 10 5rere dr de
Using the description of the solid region, set up the integral for (a) The mass(b) The center of mass(c) The moment of inertia about the z-axis.The solid in the first octant bounded by the coordinate planes and x² + y² + z² = 25 with density function ρ = kxy
The figure below shows Erie County, New York. Let ƒ(x, y)represent the total annual snowfall at the point(x, y) in the county, where R is the county.Interpret each of the following.(a)(b) Buffalo ERIE
Find the Jacobian∂(x, y)/∂(u, v) for the indicated change of variables.x = u + 3v, y = 2u - 3v
Sketch the region of integration. Then evaluate the iterated integral. ff x e-y² dy dx
Repeat Exercise 67 for a region R bounded by the graph of the equation (x - 2)² + y² = 4.Data from in Exercise 67Consider the region bounded by the graphs of y = 2, y = 4, y = x, and y = √3x and the double integral ∫R ∫ ƒ dA. Determine the limits of integration when the region R is divided
Determine whether the moment of inertia about the y-axis of the cylinder in Exercise 57 will increase or decrease for the nonconstant density ρ(x, y, z) = √x² + z² and a = 4.Data from in Exercise 57Verify the moments of inertia for the solid of uniform density. Use a computer algebra system to
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The volume of the sphere x² + y2 + z² = 1 is given by the integral v=8ff √T₁ V 0 0 √1-x² - y² dx dy.
Find the Jacobian ∂(x, y)/∂(u, v) for the indicated change of variables.x = u² + v², y = u² - y²
Sketch the region of integration. Then evaluate the iterated integral. ly sin x² dx dy
Find K such that the functionis a probability density function. f(x, y) = [ke-(x² + y²), x ≥ 0, y ≥ 0 0, elsewhere
Which of the integrals below is equal toExplain.(a)(b)(c) 3 ³², f(x, y, z) dz dy dx?
Find the Jacobian ∂(x, y)/∂(u, v) for the indicated change of variables.x = u sin θ + v cos θ, y = u cos θ + v sin θ
Sketch the region of integration. Then evaluate the iterated integral. 4 Jo Jy² √x sin x dx dy
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If ƒ(x, y) ≤ g(x, y) for all (x, y) in R, and both ƒ and g continuous are over R, then ∫R ∫ ƒ(x, y) dA ≤ ∫R ∫ ƒ(x, y) dA.
Show that the area A of the polar sector R (see figure) is A = rΔrΔθ, where r = (r1 + r₂)/2 is the average radius of R. 3 R 1₂
Use a computer algebra system to evaluate the iterated integral. 0 2x (x³ + 3y²) dy dx
Consider two solids, solid A and B solid of equal weight as shown below.(a) Because the solids have the same weight, which has the greater density? Explain.(b) Which solid has the greater moment of inertia?(c) The solids are rolled down an inclined plane. They are started at the same time and at
Use the indicated change of variables to evaluate the double integral. SS₁ X 4 3 نیا 2 1 In(x + y) dA = (u (u + v), y = /²/ (u — v - (1, 2) 1 (2, 3) R (2, 1) 2 3 (3, 2) 4 X
Find the solid region Q where the triple integralis a maximum. Use a computer algebra system to approximate the maximum value. What is the exact maximum value? JSSG₁-21² 2x² - y²-32²) dV
Solve for a in the triple integral. 0 [3-a-y² (4-x-y² ²²² 10 dz dx dy = 14 15
Use the indicated change of variables to evaluate the double integral. SS₁ R. 1 = ² (u + v), y = 1/ (v - u) 16xy dA T 5 4 3 (0, 2) 1 y (1,4) R (2, 2) (1,0) H 12 3 + 4 X
Determine the region R in the xy-plane that minimizes the value of SS G1²+ JRJ (x² + y²4) dA.
Find the Jacobian ∂(x, y)/∂(u, v) for the indicated change of variables.x = uv, y = v/u
Use a computer algebra system to evaluate the iterated integral. ff 0 2 (x + 1)(y + 1) dx dy
Evaluate Solve for a in the triple integral. lim S S · · · * cos² (275(x₂ + x₂ + · · · + x)} dx, dx₂ 11→∞0 0 2n dx, dx₂. dxn
LetFind the average value of ƒ on the interval [0, 1]. x) = f ₁² e² f(x)= et² dt.
Use a geometric argument to show that 10 10 9=x=3&dy = /9 - xº – y dx dy 97 2
Use the indicated change of variables to evaluate the double integral. Sfaxy x = u₁y = (u - v) 5 4 3 نرا 2 y (xy + x²) dA (1,3) R (1, 1) (4,4) (4,2) + + 12 3 4 5 X
Use the indicated change of variables to evaluate the double integral. SS= x = u, y = 6 5 4 3 2 1 x 1 + x²y² dA y x = 1 R V u xy=5 xy=1] x=5 4 5 6 x
Use a computer algebra system to evaluate the iterated integral. X-DJ D. a ra-x 0/ (x² + y²) dy dx
Use a computer algebra system to evaluate the iterated integral. 24-x² S.S. 0 0 exy dy dx
Evaluate dy dx, where a and b arepositive. Sofo emax{b²x², a²y²}
The following iterated integrals represent the solution to the same problem. Which iterated integral is easier to evaluate? Explain your reasoning. sin y² dy dx 10 Jx/2 2y =ff sin 10 0 sin y² dx dy
Evaluate the iterated integral. */4 ffm/4 *cos փ 10 0 cos 0 dp do do
Show that the function is a joint density function and find the required probability. 0≤x≤ 5,0 ≤ y ≤ 2 elsewhere To 10' f(x, y) 0, P(0 ≤ x ≤ 2, 1 ≤ y ≤ 2)
Find the average value of the function over the given solid. The average value of a continuous function ƒ(x, y, z) over a solid region Q iswhere V is the volume of the solid region Q.ƒ(x, y, z) = z² + 4 over the cube in the first octant bounded bythe coordinate planes and the planes x = 1, y =
Give a geometric argument for the equality. Verify the equality analytically. JJ 0 Jx 50-x² x²y² dy dx = J₁J+₁x JJ x²y² dx dy + (0, 5√2) 5 y 50-y² y=√/50-x² x²y² dx dy y = x 5 (5,5) X
Complete the iterated integrals so that each one represents the area of the region R (see figure).(a)(b) 2 1 y y=√x 1 R 2 3 y= (4,2) 2 4 X
Use a computer algebra system to approximate the iterated integral. 2 √√z²+ 4 dz dr de Jo Jo Jo v
Show that the function is a joint density function and find the required probability. faxy, 0≤x≤ 2,0 ≤ y ≤ 2 0, elsewhere P(0 ≤ x ≤ 1, 1 ≤ y ≤ 2) f(x, y) =
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If ∫R ∫ ƒ(r, θ) dA > 0, then ƒ(r, θ) > 0 for all (r, θ) in R.
Use a computer algebra system to approximate the iterated integral. T/2 m/2 cos [/²5/² * 0 0 0 p² cos 0 dp de do
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If ƒ(r, θ) is a constant function and the area of the region S is twice that of the region R, then 2 ² √ [ 1(1, 0) ds = [ S J(r₂ f(r, dA f(r. R. f(r, 0) dA.
Show that the function is a joint density function and find the required probability. f(x, y) = 7(9-x-y), 0≤ x ≤ 3, 3 ≤ y ≤ 6 elsewhere 0, P(0 ≤ x ≤ 1, 4 ≤ y ≤ 6)
Find the average value of the function over the given solid. The average value of a continuous function ƒ(x, y, z) over a solid region Q iswhere V is the volume of the solid region Q.ƒ(x, y, z) = xyz over the cube in the first octant bounded by thecoordinate planes and the planes x = 4, y = 4,
Sketch the region of integration. Then evaluate the iterated integral. S.J 0 x x√1 + y³ dy dx
Find the average value of the function over the given solid. The average value of a continuous function ƒ(x, y, z) over a solid region Q iswhere V is the volume of the solid region Q.ƒ(x, y, z) = x + y + z over the tetrahedron in the first octantwith vertices (0, 0, 0), (2, 0, 0), (0, 2, 0), and
Show that the function is a joint density function and find the required probability. f(x, y) [e-x-y, x ≥ 0, y = 0 elsewhere 0, P(0 ≤ x ≤ 1, x ≤ y ≤ 1)
Sketch the region of integration. Then evaluate the iterated integral. 2 Soft 0 3 2 + y² dy dx
Use cylindrical coordinates to find the volume of the solid bounded above by z = 8 - x² - y² and below by z = x² + y².
Find the average value of the function over the given solid. The average value of a continuous function ƒ(x, y, z) over a solid region Q iswhere V is the volume of the solid region Q.ƒ(x, y, z) = x + y over the solid bounded by the spherex² + y² + z² = 3 #fff f(x, y, z) dv
The table shows values of a function ƒ over a square region R. Divide the region into 16 equal squares and select (xi, yi) to be the point in the ith square closest to the origin. Compare this approximation with that obtained by using the point in the ith square farthest from the origin. f*f* f(x,
Sketch the region of integration. Then evaluate the iterated integral. 2 4ex² dy dx 0 J2x
Use spherical coordinates to find the volume of the solid bounded above by x² + y² + z² = 36 and below by z = √x² + y².
Consider the region bounded by the graphs of y = 2, y = 4, y = x, and y = √3x and the double integral ∫R ∫ ƒ dA. Determine the limits of integration when the region R is divided into (a) Horizontal representative elements(b) Vertical representative elements(c) Polar sectors
Define a triple integral and describe a method of evaluating a triple integral.
Use a computer algebra system to evaluate the iterated integral. 2 ff x √16x³y³ dy dx
Use a computer algebra system to evaluate the iterated integral. *2π (1+cos 0 ST. 0 6r² cos 0 dr de
Show that if λ > 1/2 there does not exist a real-valuedfunction u such that for all x in the closed interval0 ≤ x ≤ 1, u(x) = 1 + λ∫1x u(y)u( y − x) dy.
(a) Sketch the region of integration(b) Switch the order of integration(c) Use a computer algebra system to show that both orders yield the same value. 4√2y SS (x²y xy2) dx dy -
(a) Sketch the region of integration(b) Switch the order of integration(c) Use a computer algebra system to show that both orders yield the same value. 2 (4-x²/4 ST ху Jo √√√√4-x² x² + y² + 1 dy dx
Use a computer algebra system to evaluate the iterated integral. SORT. 0 T/2 (1+sin 0 150r dr de
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. a JC f(x, y) dy dx = [ c f(x, y) dx dy
Explain what is meant by an iterated integral. How is it evaluated?
Describe regions that are vertically simple and regions that are horizontally simple.
Give a geometric description of the region of integration when the inside and outside limits of integration are constants.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. x Lfs 0 0 f(x, y) dy dx - S. f. ³0 0 f(x, y) dx dy
Explain why it is sometimes an advantage to change the order of integration.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.Two different level curves of the graph of z = ƒ(x, y) canintersect.
Show that the mixed partial derivatives ƒxyy, ƒyxy, and ƒyyx are equal. f(x, y, z) = 2z x + y
Show that the mixed partial derivatives ƒxyy, ƒyxy, and ƒyyx are equal.ƒ(x, y, z) = e-x sin yz
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.A vertical line can intersect the graph of z = ƒ(x, y) at most once.
Show that thefunction satisfies Laplace's equation ∂²z/∂x² + ∂²z/∂y² = 0.z = 5xy
Show that the function satisfies the wave equation ∂²z/∂t² = c²(∂²z/∂x²).z = cos(4x + 4ct)
Show that the function satisfies Laplace's equation ∂²z/∂x² + ∂²z/∂y² = 0.z = 1/2(ey - e¯y)sin x
Show that the functionsatisfies the wave equation ∂²z/∂t² = c²(∂²z/∂x²).z = sin(x - ct)
Show that the function satisfies Laplace's equation ∂²z/∂x² + ∂²z/∂y² = 0.z = arct y/x
Show that the function satisfies Laplace's equation ∂²z/∂x² + ∂²z/∂y² = 0.z = ex sin y
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If lim f(x, y) = 0, then lim f(x, 0) = 0. (x, y) (0, 0)*
For ƒ(x, y), find all values of x and y such that ƒx(x, y) = 0 and ƒy(x, y) = 0 simultaneously. f(x, y) = 1 / / ₁ X T + + xy y
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If lim f(0, y) = 0, then lim f(x, y) = 0. (x, y)→(0, 0) (x, y)→(0, 0)*
The temperature in degrees Celsius on thesurface of a metal plate is given by 7(x, y), where x and y aremeasured in centimeters. Find the direction from point Pwhere the temperature increases most rapidly and the rate ofincrease.T(x, y) = 80 - 3x² - y², P(-1, 5)
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If ƒ is continuous for all nonzero x and y, and ƒ(0, 0) = 0, then limf(x, y) = 0. (x, y)→(0, 0)*
For ƒ(x, y), find all values of x and y such that ƒx(x, y) = 0 and ƒy(x, y) = 0 simultaneously.ƒ(x, y) = x² + xy + y² - 2x + 2y
Sketch the graph of the level surface ƒ(x, y, z) = c at the given value of c.ƒ(x, y, z) = x − y + z, c = 1
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If ƒ(x, y) = x + y, then − 1 ≤ Duƒ(x, y) ≤ 1.
For ƒ(x, y), find all values of x and y such that ƒx(x, y) = 0 and ƒy(x, y) = 0 simultaneously.ƒ(x, y) = 3x³ - 12xy + y³
Sketch the graph of the level surface ƒ(x, y, z) = c at the given value of c.ƒ(x, y, z) = 4x + y + 2z, c = 4
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If Duƒ(x, y) exists, then Duƒ(x, y) = -D-uƒ(x, y).

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