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mathematics
calculus 6th edition
Calculus 6th Edition James Stewart - Solutions
Evaluate the iterated integral. 1² S2² (x - y) dy dx J2x
If R = [–1, 3] x [0, 2], use a Riemann sum with m = 4, n = 2 to estimate the value of ∫∫ (y2 – 2x2) dA. Take the sample points to be the upper left corners of the squares.
Find the Jacobian of the transformation.x = uv, y = u/v
Plot the point whose cylindrical coordinates are given. Then find the rectangular coordinates of the point.(a) (1, π, e) (b) (1, 3π/2, 2)
Plot the point whose spherical coordinates are given. Then find the rectangular coordinates of the point.(a) (5, π, π/2) (b) (4, 3π/4, π/3)
Find ∫05f(x, y) dx and ∫01f(x, y) dy.f(x, y) = y + xey
Evaluate the iterated integral. *x+z 10 Jo Jo 6xz dy dx dz
(a) Use a Riemann sum with m = n = 2 to estimate the value of ∫∫R sin(x + y) dA, where R = [0, π] × [0, π]. Take the sample points to be lower left corners. (b) Use the Midpoint Rule to estimate the integral in part (a).
Evaluate the iterated integral. ffa (1 + 2y) dy dx Jo
Find the Jacobian of the transformation.x = e–r sin θ, y = er cos θ
Change from rectangular to cylindrical coordinates.(a) (1, –1, 4)(b) (–1, –√3, 2)
Change from rectangular to spherical coordinates.(a) (1, √3, 2√3) (b) (0, –1, –1)
Calculate the iterated integral. ²² (1 + 4xy) dx dy
Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.D = {(x, y) | 0 ≤ x ≤ 2, –1 ≤ y ≤ 1}: ρ(x, y) = xy2
Evaluate the iterated integral. 2x S² [²* f* ²2.xyz dz dy dx Jx Jo
Change from rectangular to cylindrical coordinates.(a) (2√3, 2, –1) (b) (4, –3, 2)
(a) Estimate the volume of the solid that lies below the surface z = x + 2y2 and above the rectangle R = [0, 2] × [0, 4]. Use a Riemann sum with m = n = 2 and choose the sample points to be lower right corners. (b) Use the Midpoint Rule to estimate the volume in part (a).
Evaluate the iterated integral. 2 2y $² 13³ xy dx dy Jo Jy
Evaluate the iterated integral. Jo Jo √1-2² So ze" dx dz dy
Calculate the iterated integral. 11. Jo Jo yey dx dy
Calculate the iterated integral. S² S₁² (4x³ — 9x²y³²) dy dx
Find the Jacobian of the transformation.x = es + t, y = es–t
Change from rectangular to spherical coordinates.(a) (0, √3, 1) (b) (–1, 1, √6)
Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.D = {(x, y) |0 ≤ x ≤ a,0 ≤ y ≤ b} : ρ(x, y) = cxy
Describe in words the surface whose equation is given.θ = π/4
A table of values is given for a function f(x, y) defined on R = [1, 3] x [0, 4]. (a) Estimate ∫∫R(x, y) dA using the Midpoint Rule with m = n = 2.(b) Estimate the double integral with by choosing m = n = 4 the sample points to be the points farthest from the origin.
Evaluate the iterated integral. S/² *1/2 (cos 8 or e sine dr de
Sketch the region whose area is given by the integral and evaluate the integral. 2 7 S²″ S2² r dr de
Calculate the iterated integral. 2 Jo Jo m/2 x x sin y dy dx
Calculate the iterated integral. for f* cos(x²) dy dx Jo Jo
Evaluate the iterated integral. Sze² dx dy dz
Describe in words the surface whose equation is given.r = 5
Evaluate the iterated integral. SS √₁ - v² du dv 1 Jo
Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.D is bounded by y = ex, y = 0, x = 0, and x = 1, ρ(x, y) = y
Sketch the region whose area is given by the integral and evaluate the integral. 1/2 4 cos 6 fa r dr de
Calculate the iterated integral. *1/2 5 Scos cos y dx dy J/6 J-1
Calculate the iterated integral. * Jo 10 Jx 3.xy² dy dx
Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.D is the triangular region enclosed by the lines x = 0, y = x, and 2x + y = 6: ρ(x, y) = x2
Find the Jacobian of the transformation.x = v + w2, y = w + u2, z = u + v2
Evaluate the iterated integral. π/2 y [¹² f * cos(x + y + 2) dz dx dy
Evaluate the given integral by changing to polar coordinates.∫∫D xy dA where D is the disk with center the origin and radius 3 SD xy dA, JJD
Identify the surface whose equation is given.z = 4 – r2
Evaluate the double integral. [[y²dA, D = {(x,y) | −1≤ y ≤ 1, _y_2
Calculate the iterated integral. ²² (2x + y)² dx dy Jo Jo
Identify the surface whose equation is given.ρ = sin θ sin Φ
Evaluate the iterated integral. TAXI fo* f * x² sin y dy dz dx Jo Jo Jo
The figure shows level curves of a function f in the square R = [0, 2] × [0, 2]. Use the Midpoint Rule with m = n = 2 to estimate ∫∫R f(x, y) dA. How could you improve your estimate? уч 2 1 0 А 3 1 5 6 7 2 X
Identify the surface whose equation is given.2r2 + z2 = 1
Evaluate the double integral. D - dA, D = {(x, y) | 0 ≤ x ≤ 1,0 ≤ y ≤ x²} x³ + 1
Evaluate the given integral by changing to polar coordinates.∫∫R (x + y) dA, where R is the region that lies to the left of the y-axis between the circles x2 + y2 = 1 and x2 + y2 = 4
Calculate the iterated integral. ¹²x dy dx 2 xe* Jo y
Identify the surface whose equation is given.ρ2 (sin2 Φ sin2θ + cos2Φ) = 9
Calculate the iterated integral. J J J Jo 6.xyz dz dx dy
Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.D is bounded by y = √x, y = 0, and x = 1; ρ(x, y) = x
Evaluate the triple integral.∫∫∫E 2x dV, where E = {(x, y, z) |0 ≤ y ≤ 2 0 ≤ x ≤ √4 – y2, 0 ≤ z ≤ y}
Write the equations in cylindrical coordinates.(a) z = x2 + y2(b) x2 + y2 = 2y
Evaluate the double integral. D x dA, D = {(x, y) |0 ≤ x ≤ π, 0 ≤ y ≤ sin x}
Evaluate the given integral by changing to polar coordinates.∫∫R cos(x2 + y2) dA, where R is the region that lies above the x-axis within the circle x2 + y2 = 9
Calculate the iterated integral. y [ K² ( ² + ² ) J1 J1 y dy dx
Write the equation in spherical coordinates.(a) z2 = x2 + y2 (b) x2 + z2 = 9
Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.D = {(x, y) |0 ≤ y ≤ sin(πx/L), 0 ≤ x ≤ L}; ρ(x, y) = y
Evaluate the triple integral.∫∫∫E yz cos(x5) dV, where E = {(x, y, z) |0 ≤ x ≤ 1,0 ≤ y ≤ x, x ≤ z ≤ 2x}
Evaluate the given integral by changing to polar coordinates.where R = {(x, y) x2 + y2 ≤ 4, x ≥ 0} SSR √4x² - y² dA, JJR
Write the equations in cylindrical coordinates.(a) 3x + 2y + z = 6 (b) –x2 – y2 + z2 = 1
Evaluate the double integral. ff x³dA, D = {(x,y) |1 ≤ x ≤e, 0 ≤ y ≤ lnx} D
Calculate the iterated integral. *3 Sofer ex+3y dx dy
Write the equation in spherical coordinates.(a) x2 – 2x + y2 + z2 = 0 (b) x + 2y + 3z = 1
Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ.D is bounded by the parabolas y = x2 and x = y2; ρ(x, y) = √x
Sketch the solid described by the given inequalities.0 ≤ r ≤ 2, –π/2 ≤ θ ≤ π/2, 0 ≤ z ≤ 1
Evaluate the double integral. D y²e dA, D = {(x, y) |0 ≤ y ≤ 4, 0 ≤ x ≤ y}
Evaluate the double integral by first identifying it as the volume of a solid.∫∫R 3 dA, R = {(x, y) |–2 ≤ x ≤ 2,1 ≤ y ≤ 6}
Sketch the solid described by the given inequalities.ρ ≤ 2, 0 ≤ Φ ≤ π/2, 0 ≤ θ ≤ π/2
Evaluate the triple integral.∫∫∫E y dV, where E is bounded by the planes x = 0, y = 0, z = 0, and 2x + 2y + z = 4
Evaluate the double integral by first identifying it as the volume of a solid.∫∫R (5 – x) dA, R = {(x, y) | 0 ≤ x ≤ 5, 0 ≤ y ≤ 3}
Evaluate the double integral. D dA, D= {(x, y) |0 ≤ y ≤ 1, 0≤x≤y}
Evaluate the given integral by changing to polar coordinates.∫∫R yex dA, where R is the region in the first quadrant enclosed by the circle x2 + y2 = 25
Sketch the solid described by the given inequalities.2 ≤ ρ ≤ 3, π/2 ≤ Φ ≤ π
Evaluate the triple integral.∫∫∫E x2ey dV, where E is bounded by the parabolic cylinder z = 1 – y2 and the planes z = 0, x = 1, and x = –1
Evaluate the double integral. SS x cos y dA, D is bounded by y = 0, y = x², x = 1 D
Evaluate the double integral by first identifying it as the volume of a solid.∫∫R (4 –2y) dA, R = [0, 1] × [0, 1]
Calculate the iterated integral. _S3 S² • r sin²0 de dr
Sketch the solid described by the given inequalities.ρ ≤ 1, 3π/4 ≤ Φ ≤ π
Evaluate the double integral. [[ (x + y) dA, D is bounded by y = √x and y = x²
Evaluate the triple integral.∫∫∫E xy dV, where E is bounded by the parabolic cylinders y = x2 and x = y2 and the planes z = 0 and z = x + y
Sketch the solid whose volume is given by the integral and evaluate the integral. 2 · St. Sat Str. So r dz do dr
Evaluate the triple integral.∫∫∫T x2 dV, where T is the solid tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1)
Sketch the solid whose volume is given by the integral and evaluate the integral. π/6 π/2 (3 a/6/²p² sin & dp do do 0 0
Evaluate the double integral.D is the triangular region with vertices (0, 0), (1, 2), and (0, 3) ff 2xy dA, D
Evaluate ∫∫∫E(x3 + xy2) dV, where E is the solid in the first octant that lies beneath the paraboloid z = 1 – x2 – y2.
Use a programmable calculator or computer (or the sum command on a CAS) to estimate where R = [0, 1] x [0, 1]. Use the Midpoint Rule with the following numbers of squares of equal size: 1, 4, 16, 64, 256, and 1024. R /1 + xe dA
Evaluate the double integral.D is the triangular region with vertices (0, 2), (1, 1), (3, 2) D 3 y³ dA,
Calculate the double integral. ff (6x²y³ - 5y¹) dA, R = {(x, y) | 0 ≤ x ≤ 3, 0 ≤ y ≤ 1} R
Sketch the solid whose volume is given by the integral and evaluate the integral. Өр Ар гр л 4 Jo Jo of 24-6J 2J 2/²,
Repeat Exercise 15 for the integral ∫∫R sin(x + √y) dA.Data from Exercise 15Use a programmable calculator or computer (or the sum command on a CAS) to estimate where R = [0, 1] x [0, 1]. Use the Midpoint Rule with the following numbers of squares of equal size: 1, 4, 16, 64, 256, and 1024.
Evaluate the double integral.D is enclosed by x = 0 and x = √1 – y2 D xy² dA,
Calculate the double integral. ff cos(x + 2y) dA, R = {(x, y) | 0 ≤ x ≤ m, 0 ≤ y ≤ m/2} R
If f is a constant function, f(x, y) = k, and R = [a, b] x [c, d], show that ∫∫R k dA = k(b – a)(d – c).
Use a double integral to find the area of the region.The region within both of the circles r = cos θ and r = sin θ
Sketch the solid whose volume is given by the integral and evaluate the integral. (2 10 *2 S/S² p² sin & dp do do Jπ/2 J1
Use cylindrical coordinates.Evaluate ∫∫∫E ez dV, where E is enclosed by the paraboloid z = 1 + x2 + y2, the cylinder x2 + y2 = 5, and the xy-plane.
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