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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

Sketch the domain of integration and express as an iterated integral in the opposite order. 3 V f(x,y) dx dy
Sketch the domain of integration and express as an iterated integral in the opposite order. L. "Sof 2 f(x, y) dxdy
Sketch the domain of integration and express as an iterated integral in the opposite order. S'. re ex f(x,y) dy dx
Sketch the domain D corresponding toThen change the order of integration and evaluate. So So V 4x + 5y dxdy
Change the order of integration and evaluateExplain the simplification achieved by changing the order. NT/2 . 0/ x cos(xy) dx dy
Compute the integral of ƒ(x, y) = (ln y)−1 over the domain D bounded by y = ex and y = e √x. Choose the order of integration that enables you to evaluate the integral.
Evaluate by changing the order of integration: S. Si siny dydx
Sketch the domain of integration. Then change the order of integration and evaluate. Explain the simplification achieved by changing the order. S. S y sin x x dx dy
Sketch the domain of integration. Then change the order of integration and evaluate. Explain the simplification achieved by changing the order. 2 S S x + 1 dx dy
Sketch the domain of integration. Then change the order of integration and evaluate. Explain the simplification achieved by changing the order. S. y=x xe dy dx
Sketch the domain of integration. Then change the order of integration and evaluate. Explain the simplification achieved by changing the order. 1 So So teva dy dx
Sketch the domain D where 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, and x or y is greater than 1. Then compute SS D exty dA.
Calculate where D is bounded by the lines y = x + 1, y = x, x = 0, and x = 1. SS D et dA-
Calculate the double integral of ƒ(x, y) over the triangle indicated in Figure 28.ƒ(x, y) = ex2, (A) y 4 3 2 1 54321 1 2 3 4 5 (A) 1 2 3 4 5 (C) -X -X 4 4321 y 54321 4 2 1 2 3 4 5 (B) 1 2 3 4 5 (D)
Calculate the double integral of ƒ(x, y) over the triangle indicated in Figure 28.ƒ(x, y) = 1 − 2x, (B) y 4 3 2 1 54321 1 2 3 4 5 (A) 1 2 3 4 5 (C) -X -X 4 4321 y 54321 4 2 1 2 3 4 5 (B) 1 2 3 4 5 (D)
Calculate the double integral of ƒ(x, y) over the triangle indicated in Figure 28. y 4 3 2 1 54321 1 2 3 4 5 (A) 1 2 3 4 5 (C) -X -X 4 4321 y 54321 4 2 1 2 3 4 5 (B) 1 2 3 4 5 (D)
Calculate the double integral of ƒ(x, y) over the triangle indicated in Figure 28.ƒ(x, y) = x + 1, (D) y 4 3 2 1 54321 1 2 3 4 5 (A) 1 2 3 4 5 (C) -X -X 4 4321 y 54321 4 2 1 2 3 4 5 (B) 1 2 3 4 5 (D)
Calculate the double integral of ƒ(x, y) = sin y/y over the region D in Figure 29. 2 1 y D y = x y= X
Evaluate for D in Figure 30. JJD xdA
Find the volume of the region bounded by z = 40 − 10y, z = 0, y = 0, and y = 4 − x2.
Find the volume of the region enclosed by z = 1 − y2 and z = y2 − 1 for 0 ≤ x ≤ 2.
Find the volume of the region bounded by z = 16 − y, z = y, y = x2, and y = 8 − x2.
Find the volume of the region bounded by y = 1 − x2, z = 1, y = 0, and z + y = 2.
Set up a double integral that gives the volume of the region bounded by the two paraboloids z = x2 + y2 and z = 8 − x2 − y2. (Do not evaluate the double integral.)
Compute the volume of the region bounded by z = 2 - y, z = y, x = 0, y = 0, and x + y = 1.
On April 15, the snow depth in Jocoro Provincial Park (Example 6) was given by d(x, y) = −0.01x + 0.004y + 1.3 m. Estimate the volume of the snowpack in the park in cubic meters. EXAMPLE 3 A Volume Integral Find the volume V of the region under the plane z = 2x + 3y and above the triangle D in
On May 1, there was no snow in Jocoro Provincial Park (Example 6) east of Highway 55 (i.e., for x ≥ 100). Otherwise, the snow depth was given by d(x, y) = −0.008x + 0.002y + 0.8 m. Estimate the volume of the snowpack in the park in cubic meters.
Calculate the average value of ƒ(x, y) = ex+y on the square domain [0, 1] × [0, 1].
Calculate the average y-coordinate of the points in the region given by 0 ≤ x ≤ 1, 0 ≤ y ≤ x2.
Find the average height of the “ceiling” in Figure 31 defined by z = y2 sin x for 0 ≤ x ≤ π, 0 ≤ y ≤ 1. z = y2 sin x x
Calculate the average value of the x-coordinate of a point on the domain x2 + y2 ≤ R2, x ≥ 0. What is the average value of the y-coordinate?
What is the average value of the linear function f (x, y)= mx +ny+ p ipse (2) + () =
Find the average of the square of the distance from the origin to a point in the domain D in Figure 32. 1 y 1 x = y + 1 (x, y) D 3 x
Let D be the rectangle 0 ≤ x ≤ 2, −1/8 ≤ y ≤ 1/8 , and let ƒ(x, y) = √x3 + 1. Prove that D f(x,y) dA < 32
(a) Use the inequality sin θ ≤ θ for θ ≥ 0 to show that(b) Use a computer algebra system to evaluate the double integral to three decimal places. S'S 0 sin(xy) dx dy 1 T 4
Prove the inequality ≤ π, where D is the disk x2 + y2 ≤ 4. SS D dA 4+x + y ST.
Let D be the domain bounded by y = x2 + 1 and y = 2. Prove the inequality 4 / ff (x + y ) dA 200 3 3
Let be the average of ƒ(x, y) = xy2 on D = [0, 1] × [0, 4]. Find a point P ∈ D such that ƒ(P) = (the existence of such a point is guaranteed by the Mean Value Theorem for Double Integrals). f
Verify the Mean Value Theorem for Double Integrals for f (x, y) = ex−y on the triangle bounded by y = 0, x = 1, and y = x.
Use the approximation in (12) to estimate the double integral.The following table lists the areas of the subdomains Dj of the domain D in Figure 33 and the values of a function f(x, y) at sample points P; Dj. Estimate Sf f(x, y) dA.
Use the approximation in (12) to estimate the double integral.The domain D between the circles of radii 5 and 5.2 in the first quadrant in Figure 34 is divided into six subdomains of angular width Δθ = π 1/2 , and the values of a function ƒ (x, y) at sample points are given.Compute the area of
According to Eq. (3), the area of a domain D is equal to Prove that if D is the region between two curves y = g1(x) and y = g2(x) with g2(x) ≤ g1(x) for a ≤ x ≤ b, then SS D 1 dA.
Let D be a closed connected domain and let P, Q ∈ D. The Intermediate Value Theorem (IVT) states that if ƒ is continuous on D, then ƒ(x, y) takes on every value between ƒ(P) and ƒ (Q) at some point in D.(a) Show, by constructing a counterexample, that the IVT is false if D is not
Use the fact that a continuous function on a closed bounded domain D attains both a minimum value m and a maximum value M, together with Theorem 3, to prove that the average value lies between m and M. Then use the IVT in Exercise 68 to prove the Mean Value Theorem for Double Integrals. J
Let where f (y) is a function of y alone.(a) Use the Fundamental Theorem of Calculus to prove that G"(t) = ƒ(t).(b) Show, by changing the order in the double integral, that This illustrates that the “second antiderivative” of f (y) can be expressed as a single integral. SS f(y) dy dx where
Which of (a)–(c) is not equal to 3 f(x, y, z) dzdy dx?
Which of the following is not a meaningful triple integral? (a) (b) S S J 2x+y Jo Jx+y 2x+y So Jo Jo Jx+y ex+y+z dz dy dx ex+y+z dzdy dx
Describe the projection of the region of integration W onto the xy-plane: (a) (b) S S x + y 1-x Jo Jo J f(x, y, z) dz dy dx f(x, y, z) dz dy dx
Evaluatefor the specified function ƒ and box B. SS. B f(x, y, z) dv
Evaluatefor the specified function ƒ and box B. SS. B f(x,y,z) dV
Evaluatefor the specified function ƒ and box B. SS. B f(x, y, z) dv
Evaluatefor the specified function ƒ and box B. SS. B f(x, y, z)dV
Evaluatefor the specified function ƒ and box B. SS. B f(x, y, z) dv
Evaluatefor the specified function ƒ and box B. SS. B f(x,y, z) dv
Evaluatefor the specified function ƒ and box B. SS. B f(x, y, z) dv
Evaluatefor the specified function ƒ and box B. SS. B f(x, y, z) dv
Evaluate for the function f and regionWspecified. SSS, f(x, y, z) dV W
Evaluate for the function ƒ and region Wspecified. SSS, f(x, y, z) dV W
Evaluate for the function ƒ and region Wspecified. SSS, f(x, y, z) dV W
Evaluate for the function ƒ and region Wspecified. SSS, f(x, y, z) dV W
Evaluate for the function ƒ and region Wspecified. SSS, f(x, y, z) dV W
Evaluate for the function ƒ and region Wspecified. SSS, f(x, y, z) dV W
Calculate the integral of ƒ(x, y, z) = z over the region Win Figure 11, below the hemisphere of radius 3 and lying over the triangle D in the xy-plane bounded by x = 1, y = 0, and x = y. 3 W D 1 x + y +z = 9 3 X
Calculate the integral of ƒ(x, y, z) = ez over the tetrahedron Win Figure 12 (the region in the first octant under the plane shown). x 6 Z 4
Integrate ƒ(x, y, z) = x over the region in the first octant bounded above by z = 8 − 2x2 − y2 and below by z = y2.
Compute the integral of ƒ(x, y, z) = y2 over the region within the cylinder x2 + y2 = 4, where 0 ≤ z ≤ y.
Find the triple integral of the function F(x, y, z) = z over the region in Figure 13. X 3 NA 4 1
Find the volume of the solid in R3 bounded by y = x2, x = y2, z = x + y + 5, and z = 0.
Find the volume of the solid in the first octant bounded between the planes x + y + z = 1 and x + y + 2z = 1.
Calculate where W is the region above z = x2 + y2 and below z = 5, and bounded by y = 0 and y = 1. SSS y d' dv W
Evaluate where W is the domain bounded by the elliptic cylinder and the sphere x2 + y2 + z2 = 16 in the first octant (Figure 14). SSS W xz dV,
Describe the domain of integration and evaluate: xppzpx 9-x-y2 0 0 0 TJ T 9-x
Describe the domain of integration of the following integral: 5-x-z 4-z LIES -4-z f(x, y, z) dy dx dz
Let W be the region below the paraboloid x + y =z=2 that lies above the part of the plane x + y + z = 1 in the first octant. Express SSS f(x, y, z) dv as an iterated integral (for an arbitrary function f).
Assume ƒ(x, y, z) can be expressed as a product, ƒ(x, y, z) = g(x)h(y)k(z). Show that the integral of ƒ over a box B = [a, b] × [c, d] × [p, q] can be expressed as a product of integrals as follows: SS f(x,y. 2) dv = ([^" g(x) dx) (S"h(y)dy) (Sk(2).dz) B a
Consider the integral in Example 1:Show that the integrand can be expressed as a product g(x)h(y)k(z). Then verify the equation in Exercise 27 by computing the product of integrals on the right-hand side and showing it equals the result obtained in the example.Data From Exercise 27Assume ƒ(x, y,
In Example 5, we expressed a triple integral as an iterated integral in the three ordersExample 5 dzdy dx, Write this integral in the three other orders: dxdz dy, dxdz dy, and dy dz dx dzdxdy, dx dy dz, and dy dx dz
Let W be the region shown in Figure 15, bounded by y +z = 2, 2x = y, x = 0, and z = 0
Let(see Figure 16). Express as an iterated integral in the order dz dy dx (for an arbitrary function ƒ). W = {(x, y, z): x + y z
Repeat Exercise 31 for the order dx dy dz.Data From Exercise 31Let(see Figure 16). Express as an iterated integral in the order dz dy dx (for an arbitrary function ƒ). W = {(x, y, z): x + y z
Let W be the region bounded by z = 1 − y2, y = x2, and the plane z = 0. Calculate the volume of Was a triple integral in the order dz dy dx.
Calculate the volume of the region W in Exercise 33 as a triple integral in the following orders:(a) dx dz dy (b) dy dz dxData From Exercise 33Let W be the region bounded by z = 1 − y2, y = x2, and the plane z = 0. Calculate the volume of Was a triple integral in the order dz dy dx.
Draw the region W and then set up but do not compute a single triple integral that yields the volume of W.The region W is bounded by the surfaces given by z = 1 − y2, x = 0 and z = 0, z + x = 3.
Draw the region W and then set up but do not compute a single triple integral that yields the volume of W.The region W is bounded by the surfaces given by z = x2, z + y = 1, and z − y = 1.
Draw the region W and then set up but do not compute a single triple integral that yields the volume of W.The region W is bounded by the surfaces given by z = y2, y = z2 and x = 0, x + y + z = 4.
Draw the region W and then set up but do not compute a single triple integral that yields the volume of W.The region W is underneath z = 1 − x2 and also bounded by y = 0, z = 0, and y = 3 − x2 − z2.
Compute the average value of ƒ(x, y, z) over the region W. f(x, y, z) = xy sin(az); W = [0, 1] [0, 1] [0, 1]
Compute the average value of ƒ(x, y, z) over the region W. f(x, y, z)= xyz; W: 0z y x 1
Compute the average value of ƒ(x, y, z) over the region W. f(x, y, z)= e; W:0 y 1-x, 0zx
Compute the average value of ƒ(x, y, z) over the region W.W bounded by the planes 2y + z = 1, x = 0, x = 1, z = 0, and y = 0 f(x,y,z) = x + y + z;
Let be the Riemann sum approximation -I = SSS 0 f(x, y, z) dV and let SN.N.N
Let be the Riemann sum approximation -I = SSS 0 f(x, y, z) dV and let SN.N.N
Use Integration by Parts to verify Eq. (9). Cn (n) = Cn-2
Compute the volume An of the unit ball in Rn for n = 8, 9, 10. Show that Cn ≤ 1 for n ≥ 6 and use this to prove that of all unit balls, the five-dimensional ball has the largest volume. Can you explain why An tends to 0 as n→∞?
Which of the following represent the integral of ƒ(x, y) = x2 + y2 over the unit circle? (a) (c) -2x S' S, ~2 Loft r dr de r dr de (b) (d) -2r r dr de S S LC'Paras 25 r dr de
What are the limits of integration in if the integration extends over the following regions?(a) x2 + y2 ≤ 4, −1 ≤ z ≤ 2(b) Lower hemisphere of the sphere of radius 2, center at origin SSS f(r, 0,z)r dr do dz
What are the limits of integration inif the integration extends over the following spherical regions centered at the origin?(a) Sphere of radius 4(b) Region between the spheres of radii 4 and 5(c) Lower hemisphere of the sphere of radius 2 SSS f(p.6.0) p sin o dp do de
An ordinary rectangle of sides Δx and Δy has area Δx Δy, no matter where it is located in the plane. However, the area of a polar rectangle of sides Δr and Δθ depends on its distance from the origin. How is this difference reflected in the Change of Variables Formula for polar coordinates?
The volume of a sphere of radius 3 is 36π. What is the value of each of the following integrals? (a) (b) (c) (d) So f f psinedpdo p sin o dp do de 2r r/2 * p sin o dp do do 0 /4 S* * p sing 0 /4 - -/4 p sin o dp do de p sin o dp do de
Find the dimensions of the box of maximum volume with its sides parallel to the coordinate planes that can be inscribed in the ellipsoid (Figure 6) a + + NI = 1

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