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mathematics
calculus early transcendentals 9th

Calculus Early Transcendentals 9th Edition James Stewart, Daniel K. Clegg, Saleem Watson, Lothar Redlin - Solutions

Use a computer algebra system to find the exact area of the surfacez = 1 + 2x + 3y + 4y2, 1 ≤ x ≤ 4, 0 ≤ y ≤ 1.
Use a double integral to find the area of the region D. r= sin 0 r= cos 0
Calculate the iterated integral. dy dx Ji xy
Use a computer algebra system to find the exact area of the surfacez = 1 + x + y + x2 –2 ≤ x ≤ 1 –1 ≤ y ≤ 1Illustrate by graphing the surface.
Use a double integral to find the area of the region D. r= 1/2 D r2= cos 20
Calculate the value of the multiple integral. Sle ye dA, where R = {(x, y) | 0
Calculate the iterated integral. dy dx y
(a) Express the triple integral ∫∫∫E f(x, y, z) dV as an iterated integral in spherical coordinates for the given function f and solid region E.(b) Evaluate the iterated integral. f(x, y, z) = Vx? + y² + z²
Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why it’s easier. || sin'x dA, D D is bounded by y cos x, 0
Use the given transformation to evaluate the integral.∫∫R xy dA, where R is the region in the first quadrant bounded by the lines y = x and y = 3x and the hyperbolas xy = 1, xy = 3; x = u/v, y = v
Use a double integral to find the area of the region D.D is the loop of the rose r = sin 3θ in the first quadrant.
Calculate the iterated integral. C ve* dx dy Jo Jo
(a) Express the triple integral ∫∫∫E f(x, y, z) dV as an iterated integral in spherical coordinates for the given function f and solid region E.(b) Evaluate the iterated integral.f(x, y, z) = xy ZA x²+ y? +z?= 8 E y
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 is bounded by y = x3, y = 2x + 4, x = 0 6x2 dA, D
Use the given transformation to evaluate the integral.∫∫R y2 dA, where R is the region bounded by the curves xy = 1, xy = 2, xy2 = 1, xy2 = 2; u = xy, v = xy2. Illustrate by using a graphing calculator or computer to draw R.
Use a computer algebra system to find, correct 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.
Calculate the iterated integral. T/2 I* * sin'o dộ dt
A lamina with constant density ρ(x, y) = ρ occupies the given region. Find the moments of inertia Ix and Iy and the radii of gyration x̅̅ and y̅̅.The rectangle 0 ≤ x ≤ b, 0 ≤ y ≤ h
Evaluate the double integral. x cos y dA, Dis bounded by y = 0, y = x', x = 1
(a) Set up an iterated integral in polar coordinates for the volume of the solid under the surface and above the region D.(b) Evaluate the iterated integral to find the volume of the solid. ZA z=1+xy D ²+y²= 4, z = 0
Calculate the iterated integral. xy/x? + y? dy dx Vx²
A lamina with constant density ρ(x, y) = ρ occupies the given region. Find the moments of inertia Ix and Iy and the radii of gyration x̅̅ and y̅̅.The triangle with vertices (0, 0), (b, 0), and (0, h)
Use spherical coordinates.Evaluate ∫∫∫E y2z2 dV, where E lies above the cone Φ = π/3 and below the sphere ρ = 1.
Evaluate the double integral. (x² + 2y) dA, D is bounded by y = x, y = x', x > 0
(a) Set up an iterated integral in polar coordinates for the volume of the solid under the surface and above the region D.(b) Evaluate the iterated integral to find the volume of the solid. z= x? + y? ZA +y²= 1, 2% 0 D y x²+ y² = 4, z = 0
(a) Find a function f such that F = ∇f(b) Use part (a) to evaluate ∫C F · dr along the given curve C.F(x, y) = (2x, 4y), C is the arc of the parabola x = y2 from (4, −2) to (1, 1).
Calculate the value of the multiple integral.∫∫D y dA, where D is the region in the first quadrant bounded by the parabolas x = y2 and x = 8 – y2
Calculate the iterated integral. S v(u + v?)* du dv
A lamina with constant density ρ(x, y) = ρ occupies the given region. Find the moments of inertia Ix and Iy and the radii of gyration x̅̅ and y̅̅.The part of the disk x2 + y2 ≤ a2 in the first quadrant
Calculate the iterated integral. + ids ip sp 1 + s^ TI
Let f be continuous on [0, 1] and let R be the triangular region with vertices (0, 0), (1, 0), and (0, 1). Show that f(x + y) dA = uf (u) du R
Sketch the solid whose volume is given by the iterated integral. ri (1-x (2-2z dy dz dx Jo Jo
The figure shows a surface and a rectangle R in the xy-plane.(a) Set up an iterated integral for the volume of the solid that lies under the surface and above R.(b) Evaluate the iterated integral to find the volume of the solid. z = x² + y? ZA 3 2. y R-
Use geometry or symmetry, or both, to evaluate the double integral.D = {(x, y) | |x| + |y| ≤ 1} (2 + x*y³ - y sin x) dA, D
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y) = i + 1/2 j y F (0, 3) F (2, 2) 2+ F (1, 0) -2 FIGURE 5 F(x, y) =-yi+xj 2.
The figure shows a curve C and a contour map of a function f whose gradient is continuous. Find ∫C F · dr. y 09, 50 40 30 20 10
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y) = 2 i − j y F (0, 3) F (2, 2) 2+ F (1, 0) -2 FIGURE 5 F(x, y) =-yi+xj 2.
Let C be a simple closed piecewise-smooth space curve that lies in a plane with unit normal vector n = (a, b, c) and has positive orientation with respect to n. Show that the plane area enclosed by C is (bz - cy) dx + (cx - az) dy + (ay – bx) dz
Find(a) The curl(b) The divergence of the vector field. F(x, y, z) = i + 1 + z j + k 1+ x 1+ y
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f.F(x, y) = y2exy i + (1 + xy)exy j
Evaluate the surface integral.∫∫S (x + y + z) dS, S is the parallelogram with parametric equations x = u + v, y = u − v, z = 1 + 2u 1 v, 0 < u < 2, 0 < v < 1 ZA
Prove the following identity: V(F · G) = (F V)G + (G· V)F + FX curl G + G X curl F
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.∫C ln(xy) dx + (y/x) dy, C is the rectangle with vertices (1, 1), (1, 4), (2, 4), and (2, 1)
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y) = y i + (x + y) j y F (0, 3) F (2, 2) 2+ F (1, 0) -2 FIGURE 5 F(x, y) =-yi+xj 2.
Find(a) The curl(b) The divergence of the vector field.F(x, y, z) = ln(2y + 3z) i + ln(x + 3z) j + ln(x + 2y) k
Evaluate the line integral, where C is the given plane curve.∫C ex dx, C is the arc of the curve x = y3 from (−1, −1) to (1, 1)
(a) What does it mean to say that ∫C F · dr is independent of path?(b) If you know that ∫C F · dr is independent of path, what can you say about F?
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. Le f(x, y) ds = -fc f(x, y) ds -Sc f(x, y) ds %3D
Evaluate the surface integral. l xyz ds, S is the cone with parametric equations x = u cos v, y = u sin v, z = u, 0 < u
Use Green’s Theorem to evaluate the line integral along the given positively oriented curve.∫C x2y2 dx + y tan−1y dy, C is the triangle with vertices (0, 0), (1, 0), and (1, 3).
Find(a) The curl(b) The divergence of the vector field.F(x, y, z) = (ex sin y, ey sin z, ez sin x)
Evaluate the line integral, where C is the given plane curve.∫C (x + 2y) dx + x2 dy (2, 1) C (3, 0)
Use a computer to graph the parametric surface. Indicate on the graph which grid curves have u constant and which have v constant. r(и, о) — (и?, 0?, и + о). -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.If F and G are vector fields and divF = divG, then F = G.
Evaluate the surface integral. ls y ds, S is the helicoid with vector equation r(u, v) = (u cos v, u sin v, v), 0 < u < 1,0
The set of all points within a perpendicular distance r from a smooth simple curve C in R3 form a “tube,” which we denote by Tube(C, r); see the figure at the left. (We assume that r is small enough that the tube does not intersect itself.) It may seem that the volume of such a tube would
Find(a) The curl(b) The divergence of the vector field.F(x, y, z) = (arctan(xy), arctan(yz), arctan(zx))
Use a computer to graph the parametric surface. Indicate on the graph which grid curves have u constant and which have v constant. r(и, о) — (и, о*, — ), -v), -2 < u < 2, -2 < v
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.The work done by a conservative force field in moving a particle around a closed path is zero.
Evaluate the surface integral. Is (x? + y?) dS, S is the surface with vector equation r(u, v) = (2uv, ư – v², u² + v?), u? + v? < 1
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y, z) = i y F (0, 3) F (2, 2) 2+ F (1, 0) -2 FIGURE 5 F(x, y) =-yi+xj 2.
Evaluate the line integral, where C is the given space curve. Sex'y ds, C: x= cos t, y = sin t, z = t, 0
Use a computer to graph the parametric surface. Indicate on the graph which grid curves have u constant and which have v constant. г(и, v) — (из, и sin v, и сos v), -1 < u
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.If F and G are vector fields, thencurl(F + G) = curl F + curl G
Evaluate the surface integral. ls x'yz ds, S is the part of the plane z = 1 + 2x + 3y that lies above the rectangle [0, 3] X [0, 2]
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y, z) = z i y F (0, 3) F (2, 2) 2+ F (1, 0) -2 FIGURE 5 F(x, y) =-yi+xj 2.
Use a computer to graph the parametric surface. Indicate on the graph which grid curves have u constant and which have v constant. г(и, v) — (и, sin(и + v), sin v), ーTSUST,-TミvミT
Find the work done by the force fieldF(x, y, z) = z i + x j + y kin moving a particle from the point (3, 0, 0) to the point (0, π/2, 3) along each path.(a) A straight line(b) The helix x = 3 cos t, y = t, z = 3 sin t
Evaluate the surface integral. Sls xz dS, S is the part of the plane 2x + 2y + z = 4 that lies in the first octant
Use Stokes’ Theorem to evaluate ∫C F · dr. In each case C is oriented counterclockwise as viewed from above, unless otherwise stated.F(x, y, z) = (−yx2, xy2, exy), C is the circle in the xy-plane of radius 2 centered at the origin.
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y, z) = −y i y F (0, 3) F (2, 2) 2+ F (1, 0) -2 FIGURE 5 F(x, y) =-yi+xj 2.
Use a computer to graph the parametric surface. Indicate on the graph which grid curves have u constant and which have v constant. sin v, y = cos u sin 4v, z = sin 2u sin 4v, 0
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.If S is a sphere and F is a constant vector field, then ∫∫S F · dS = 0.
The figure shows the vector field F(x, y) = (2xy, x2) and three curves that start at (1, 2) and end at (3, 2).(a) Explain why ∫C F · dr has the same value for all three curves.(b) What is this common value? y. 3+ 2+ 1 3. 2.
Evaluate the surface integral. Slx ds, S is the triangular region with vertices (1, 0, 0), (0, -2, 0), and (0, 0, 4)
Use Stokes’ Theorem to evaluate ∫C F · dr. In each case C is oriented counterclockwise as viewed from above, unless otherwise stated.F(x, y, z) = zex i + (z − y3) j + (x − z3) k, C is the circle y2 + z2 = 4, x = 3, oriented clockwise as viewed from the origin.
Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.F(x, y, z) = i + k y F (0, 3) F (2, 2) 2+ F (1, 0) -2 FIGURE 5 F(x, y) =-yi+xj 2.
Evaluate the line integral, where C is the given space curve. Se(x? + y? + z*) ds, C: x = t, y = cos 2t, z = sin 2t, 0
Use a computer to graph the parametric surface. Indicate on the graph which grid curves have u constant and which have v constant. x = cos u, y = sin u sin v, 0
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.There is a vector field F such thatcurl F = x i + y j + z k
Evaluate ∫C F · dr for the vector field F(x, y) = 2xy i + (x2 + sin y) j and the curve C shown.a.b. yA (2, п/2) C
Use Stokes’ Theorem to evaluate ∫C F · dr. In each case C is oriented counterclockwise as viewed from above, unless otherwise stated.F(x, y, z) = x2y i + x3 j + ez tan−1z k, C is the curve with parametric equations x = cos t, y = sin t, z = sin t, 0 ≤ t ≤ 2π. ZA (-1,0, 0) y (1, 0, 0)
Use Green’s Theorem to evaluate ∫C F · dr. (Check the orientation of the curve before applying the theorem.)∫C (3 + ex2) dx + (tan−1y + 3x2) dy у. +y? = 4 1 C x2+ y? = 1 1 2.
Evaluate the line integral, where C is the given space curve. Sexye" dy, C: x = t, y = t, z = t, 0
Evaluate the surface integral. l, z? dS, S is the part of the paraboloid x = y² + z? given by 0
Let F(x, y) = (3x2 + y2) i + 2xy j and let C be the curve shown.(a) Evaluate ∫C F · dr directly.(b) Show that F is conservative and find a function f such that F = ∇f .(c) Evaluate ∫C F · dr using Theorem 2.(d) Evaluate ∫C F · dr by first replacing C by a simpler curve that has the same
Use Stokes’ Theorem to evaluate ∫C F · dr. In each case C is oriented counterclockwise as viewed from above, unless otherwise stated.F(x, y, z) = (x3 − z, xy, y + z 2), C is the curve of intersection of the paraboloid z = x2 + y2 and the plane z = x ZA z=x²+y? z=x'
Use the Divergence Theorem to calculate the surface integral ∫∫S F · dS; that is, calculate the flux of F across S.F(x, y, z) = (xy − z2) i + x3 √z j + (xy + z2) k, S is the surface of the solid bounded by the cylinder x = y2 and the planes x + z = 1 and z = 0. ZA x+z=1 r= y? y
Use Green’s Theorem to evaluate ∫C F · dr. (Check the orientation of the curve before applying the theorem.)∫C (x2/3 + y2) dx + (y4/3 − x2) dy y (4, 2) x= y? C (4, 0)
Evaluate the line integral, where C is the given space curve. Sc ye dz + x In x dy – y dx, C: x = e', y -2t, z = In t, 1
Evaluate the surface integral. ls y'z² dS, S is the part of the cone y = Vx? + z? given by 0 < y < 5 2
A vector field F and a curve C are given.(a) Show that F is conservative and find a potential function f.(b) Evaluate ∫C F · dr using Theorem 2.(c) Evaluate ∫C F · dr by first replacing C with a line segment that has the same initial and terminal points. F(x, y) = (sin y + e", x cos y), C: x
Evaluate the line integral, where C is the given space curve. Sez dx + xy dy + y² dz, C: x = sin t, y = cos t, z = tan t, -T/4
Evaluate the surface integral. ls x ds, S is the surface y = x? + 4z, 0
A vector field F and a curve C are given.(a) Show that F is conservative and find a potential function f.(b) Evaluate ∫C F · dr using Theorem 2.(c) Evaluate ∫C F · dr by first replacing C with a line segment that has the same initial and terminal points. F(x, y) = (ye", xe"), C: x= sin t, y =
Evaluate the line integral, where C is the given space curve. Scy dx + z dy + x dz, C: x = Vi, y= t, z= t, 1
Evaluate the surface integral. Sls y² dS, S is the part of the sphere x2 + y2 + z? = 1 that lies above %3| the cone z = Vx2 + y?
Evaluate ∫C ∇f · dr, where f(x, y, z) = xy2z + x2 and C is the curve x = t2, y = et2−1, z = t2 + t, −1 ≤ t ≤ 1.
Use the Divergence Theorem to calculate the surface integral ∫∫S F · dS; that is, calculate the flux of F across S.F = |r| r, where r = x i + y j + z k, S consists of the hemisphere z = √1 − x2 − y2 and the disk x2 + y2 ≤ 1 in the xy-plane.
Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = ∇f.F(x, y, z) = (ln y, (x/y) + ln z, y/z)

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