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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Let SN,N be the Riemann sum for cos(xy) dx dy using midpoints as sample points.(a) Calculate S4,4.(b) Use a computer algebra system to calculate SN,N for N = 10, 50, 100. S S
Let D be the shaded domain in Figure 1. y 2 1.5 1 0.5 0 O O 0 0.5 0 1 1.5 0 -D 2
Explain the following: (a) (b) LL sin(xy) dxdy = 0 LL cos(xy) dx dy> 0
Sketch the domain D and calculate D = {0 ≤ x ≤ 4, 0 ≤ y ≤ x}, ƒ(x, y) = cos y SS D f(x, y) dA.
Sketch the domain D and calculate D = {0 ≤ x ≤ 2, 0 ≤ y ≤ 2x − x2}, ƒ(x, y) = √xy SS D f (x,y) dA.
Sketch the domain D and calculate D = {0 ≤ x ≤ 1, 1 − x ≤ y ≤ 2 − x}, ƒ(x, y) = ex + 2y SS D f(x, y) dA.
Sketch the domain D and calculate D = {1 ≤ x ≤ 2, 0 ≤ y ≤ 1/x}, ƒ(x, y) = cos(xy) SS D f(x, y) dA.
Sketch the domain D and calculate D = {0 ≤ y ≤ 1, 0.5y2 ≤ x ≤ y2}, ƒ(x, y) = ye1 + x SS D f(x,y) dA.
Sketch the domain D and calculate D = {1 ≤ y ≤ e, y ≤ x ≤ 2y}, ƒ(x, y) = ln(x + y) SS D f(x, y) dA.
Express as an iterated integral in the order dx dy. 9-1 LL -3 0 f(x,y) dy dx
Let W be the region bounded by the planes y = z, 2y + z = 3, and z = 0 for 0 ≤ x ≤ 4. (a) Express the triple integral (x, y,2)dV as an iterated integral in the order dy dx dz (project W onto the
Let D be the domain between y = x and y = √x. Calculate as an iterated integral in the order dx dy and dy dx. SS xydA
Find the double integral of ƒ(x, y) = x3y over the region between the curves y = x2 and y = x(1 − x).
Change the order of integration and evaluate xdx dy (x + y)/2 S S 0 0
Verify directly that S S ***, -f 14 S dy dx 1 + x -y dx dy 1 + x -y
Rewrite by interchanging the order of integration, and evaluate. So I'VE ydx dy -- (1 + x + y)
Use cylindrical coordinates to compute the volume of the region defined by 4 − x2 − y2 ≤ z ≤ 10 − 4x2 − 4y2.
Evaluate where D is the shaded domain in Figure 2. Sb D xdA,
Find the volume of the region between the graph of the function ƒ(x, y) = 1 − (x2 + y2) and the xy-plane.
Calculate B = {0 ≤ x ≤ 2, 0 ≤ y ≤ 1, 1 ≤ z ≤ 3} as an iterated integral in two different ways. SSS (xy+z) dV, where B
Calculate W = {0 ≤ x ≤ 1, x ≤ y ≤ 1, x ≤ z ≤ x + y} SSSW xyzdv, where
Describe a region whose volume is equal to: (a) (b) (c) 25 r/2 S S S S -2 J/3 -27 3 S S L p sin o dp do de r dr do dz 9-1 rdz dr de
Evaluate (x3 + y2 + z) dx dy dz. JJJ (x + y +z) dx dydz.
Evaluate (x + y + z) dz dy dx. I = -1 JO 1-x S 0
Find the volume of the solid contained in the cylinder x2 + y2 = 1 below the surface z = (x + y)2 and above the surface z = −(x − y)2.
Use polar coordinates to calculate whereDis the region in the first quadrant bounded by the spiral r = θ, the circle r = 1, and the x-axis. SS x + y dA, D
Calculate whereD = {π/2 ≤ x2 + y2 ≤ π} S sin(x + y) dA,
Express in cylindrical coordinates and evaluate: 0 /1-x x+y 0 z dzdy dx
Use spherical coordinates to calculate the triple integral of ƒ(x, y, z) = x2 + y2 + z2 over the region 1x + y+2 4
Convert to spherical coordinates and evaluate: -4-x Jo 4-x-y e-(x+y +2/dz dy dx
Find the average value of ƒ(x, y, z) = xy2z3 on the box [0, 1] × [0, 2] × [0, 3].
LetW be the ball of radius R in R3 centered at the origin, and let P = (0, 0, R) be the North Pole. Let dP(x, y, z) be the distance from P to (x, y, z). Show that the average value of dP over the
Express the average value of ƒ(x, y) = exy over the ellipse x2/2 + y2 = 1 as an iterated integral, and evaluate numerically using a computer algebra system.
Use cylindrical coordinates to find the mass of the solid bounded by z = 8 − x2 − y2 and z = x2 + y2, assuming a mass density of ƒ(x, y, z) = (x2 + y2)1/2.
Let W be the portion of the half-cylinder x2 + y2 ≤ 4, x ≥ 0 such that 0 ≤ z ≤ 3y. Use cylindrical coordinates to compute the mass of W if the mass density is ρ(x, y, z) = z2.
Use cylindrical coordinates to find the mass of a cylinder of radius 4 and height 10 if the mass density at a point is equal to the square of the distance from the cylinder’s central axis.
Find the centroid of the region W bounded, in spherical coordinates, by ϕ = ϕ0 and the sphere ρ = R.
Using cylindrical coordinates, prove that the centroid of a right circular cone of height h and radius R is located at height h/4 on the central axis.
Find the centroid of solid (A) in Figure 4 defined by x2 + y2 ≤ R2, 0 ≤ z ≤ H, and π/6 ≤ θ ≤ 2π, where θ is the polar angle of (x, y).
Calculate the total charge on a plate D in the shape of the ellipse with the polar equationwith the disk x2 + y2 ≤ 1 removed (Figure 11) assuming a charge density of ρ(r, θ) = 3r−4 C/cm2. 1 sin
Find the centroid of the given region assuming the density δ(x, y) = 1.Quarter circle x2 + y2 ≤ R2, x ≥ 0, y ≥ 0
Find the centroid of the given region assuming the density δ(x, y) = 1.Lamina bounded by the x- and y-axes, the line x = M, and the graph of y = e−x
Show that the \(z\)-coordinate of the centroid of the tetrahedron bounded by the coordinate planes and the plane\[\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\]in Figure 14 is \(\bar{z}=c / 4\). Conclude by
Find the centroid of the region \(\mathcal{W}\) in Figure 15 , lying above the sphere \(x^{2}+y^{2}+z^{2}=6\) and below the paraboloid \(z=4-x^{2}-y^{2}\). 4 2 M 2=4 x 2 x + y +2=6 6
Let \(R>0\) and \(H>0\), and let \(\mathcal{W}\) be the upper half of the ellipsoid \(x^{2}+y^{2}+(R z / H)^{2}=R^{2}\), where \(z \geq 0\) (Figure 16). Find the centroid of \(\mathcal{W}\) and
Find the center of mass of the region with the given mass density \(\delta\).Region bounded by \(y=6-x, x=0, y=0 ; \quad \delta(x, y)=x^{2}\)
Find the center of mass of the region with the given mass density \(\delta\).Region bounded by \(y^{2}=x+4\) and \(x=0 ; \quad \delta(x, y)=|y|\)
Find the center of mass of the region with the given mass density \(\delta\).Region \(|x|+|y| \leq 1 ; \quad \delta(x, y)=(x+1)(y+1)\)
Find the center of mass of the region with the given mass density δδ.Semicircle x2+y2≤R2,y≥0;δ(x,y)=yx2+y2≤R2,y≥0;δ(x,y)=y
Find the zz-coordinate of the center of mass of the first octant of the unit sphere with mass density δ(x,y,z)=yδ(x,y,z)=y (Figure 17). 1
Find the center of mass of a cylinder of radius 2 and height 4 and mass density \(e^{-z}\), where \(z\) is the height above the base.
Let R be the rectangle [−a, a] × [b, −b] with uniform density and total mass M. Calculate:(a) The mass density δ of R(b) Ix and I0(c) The radius of gyration about the x-axis
Calculate \(I_{0}\) and \(I_{x}\) for the disk \(\mathcal{D}\) defined by \(x^{2}+y^{2} \leq 16\) (in meters), with total mass \(1000 \mathrm{~kg}\) and uniform mass density. Hint: Calculate
Let DD be the triangular domain bounded by the coordinate axes and the line y=3−xy=3−x, with mass density δ(x,y)=yδ(x,y)=y. Compute the given quantities.Total mass
Let \(\mathcal{D}\) be the triangular domain bounded by the coordinate axes and the line \(y=3-x\), with mass density \(\delta(x, y)=y\). Compute the given quantities.Center of mass
Let DD be the triangular domain bounded by the coordinate axes and the line y=3−xy=3−x, with mass density δ(x,y)=yδ(x,y)=y. Compute the given quantities.IxIx
Let DD be the triangular domain bounded by the coordinate axes and the line y=3−xy=3−x, with mass density δ(x,y)=yδ(x,y)=y. Compute the given quantities.I0I0
Let DD be the domain between the line y=bx/ay=bx/a and the parabola y=bx2/a2y=bx2/a2, where a,b>0a,b>0. Assume the mass density is δ(x,y)=1δ(x,y)=1 for Exercise 37 and δ(x,y)=xyδ(x,y)=xy for
Let DD be the domain between the line y=bx/ay=bx/a and the parabola y=bx2/a2y=bx2/a2, where a,b>0a,b>0. Assume the mass density is δ(x,y)=1δ(x,y)=1 for Exercise 37 and δ(x,y)=xyδ(x,y)=xy for
Let DD be the domain between the line y=bx/ay=bx/a and the parabola y=bx2/a2y=bx2/a2, where a,b>0a,b>0. Assume the mass density is δ(x,y)=1δ(x,y)=1 for Exercise 37 and δ(x,y)=xyδ(x,y)=xy for
Let DD be the domain between the line y=bx/ay=bx/a and the parabola y=bx2/a2y=bx2/a2, where a,b>0a,b>0. Assume the mass density is δ(x,y)=1δ(x,y)=1 for Exercise 37 and δ(x,y)=xyδ(x,y)=xy for
Calculate the moment of inertia IxIx of the disk DD defined by x2+y2≤R2x2+y2≤R2 (in meters), with total mass MM kilograms. How much kinetic energy (in joules) is required to rotate the disk about
Calculate the moment of inertia IzIz of the box W=[−a,a]×[−a,a]×[0,H]W=[−a,a]×[−a,a]×[0,H] assuming that WW has total mass MM.
Show that the moment of inertia of a sphere of radius RR of total mass MM with uniform mass density about any axis passing through the center of the sphere is 25MR225MR2. Note that the mass density
Sketch the region of integration and evaluate by changing to polar coordinates. -2x-x S S 1 x + y 2 dy dx
Use cylindrical coordinates to calculate the integral of the function ƒ(x, y, z) = z over the region above the disk x2 + y2 ≤ 1 in the xy-plane and below the surface z = 4 + x2 + y2.
Use cylindrical coordinates to calculate for the given function and region. SSS f(x,y,z) dv
Use cylindrical coordinates to calculate for the given function and region. SSS f(x,y,z) dv
Use cylindrical coordinates to calculate for the given function and region. SSS f(x,y,z) dv
Use cylindrical coordinates to calculate for the given function and region. SSS f(x,y,z) dv
Use cylindrical coordinates to calculate for the given function and region. SSS f(x, y, z) dv
Use cylindrical coordinates to calculate for the given function and region. SSS f(x, y,z) dv
Express the triple integral in cylindrical coordinates. S. Jy=1-x Jz=0 V= 1-x f(x, y, z) dzdy dx
Express the triple integral in cylindrical coordinates. y=1-x LLL -1 Jy=0 =0 f(x,y,z) dzdy dx
Express the triple integral in cylindrical coordinates. S.S. Jy=0 =1-x +y Jz=0 f(x,y, z) dz dy dx
Express the triple integral in cylindrical coordinates. =2x-1 x + y JJ J z=0 f(x, y, z) dz dy dx
Find the equation of the right-circular cone in Figure 22 in cylindrical coordinates and compute its volume. X N R H
Use cylindrical coordinates to integrate ƒx, y, z) = z over the intersection of the solid hemisphere x2 + y2 + z2 ≤ 4, z ≥ 0, and the cylinder x2 + y2 =≤ 1.
Find the volume of the region appearing between the two surfaces in Figure 23. X Z 8 z = x + y z=8-x-y2 y
Use cylindrical coordinates to find the volume of a sphere of radius 2a from which a central cylinder of radius a has been removed.
Use cylindrical coordinates to show that the volume of a sphere of radius a from which a central cylinder of radius b has been removed, where 0 < b < a, only depends on the height of the band that
Use cylindrical coordinates to find the volume of the region bounded below by the plane z = 1 and above by the sphere x2 + y2 + z2 = 4.
Use spherical coordinates to find the volume of the region bounded below by the plane z = 1 and above by the sphere x2 + y2 + z2 = 4.
Use spherical coordinates to find the volume of a sphere of radius 2 from which a central cylinder of radius 1 has been removed.
Use spherical coordinates to calculate the triple integral of ƒ(x, y, z) over the given region. f(x, y, z) = y; x + y+z 1, x,y,z 0
Use spherical coordinates to calculate the triple integral of ƒ(x, y, z) over the given region. f(x,y, z)= 1 x + y+z 5 x + y + z 25
Use spherical coordinates to calculate the triple integral of ƒ(x, y, z) over the given region. f(x,y, z)= x + y; p1
Find the total mass of the rectangle 0 ≤ x ≤ 1, 0 ≤ y ≤ 2 assuming a mass density of 8(x, y) = 2x + y
Calculate the total mass of a plate bounded by y = 0 and y = x−1 for 1 ≤ x ≤ 4 (in meters) assuming a mass density of δ(x, y) = y/x kg/m2.
Find the total charge in the region under the graph of y = 4e−x2/2 for 0 ≤ x ≤ 10 (in centimeters) assuming a charge density of δ(x, y) = 10−6 xy coulombs per square centimeter (C/cm2).
Find the total population within a 4-km radius of the city center (located at the origin) assuming a population density of δ(x, y) = 2000(x2 + y2)−0.2 people per square kilometer.
Find the total population within the sector 2|x| ≤ y ≤ 8 assuming a population density of δ(x, y) = 100e−0.1y people per square kilometer.
Find the total mass of the solid region W defined by x ≥ 0, y ≥ 0, x2 + y2 ≤ 4, and x ≤ z ≤ 32 − x (in centimeters) assuming a mass density of δ(x, y, z) = 6y g/cm3.
Calculate the total charge of the solid ball x2 + y2 + z2 ≤ 5 (in centimeters) assuming a charge density (in coulombs per cubic centimeter) of 8(x, y, z) = (3.10-8)(x + y + z)/2
Compute the total mass of the plate in Figure 10 assuming amass density of ƒ(x, y) = x2/(x2 + y2) g/cm2. INT 3 10
Assume that the density of the atmosphere as a function of altitude h (in kilometers) above sea level is δ(h) = ae−bh kg/km3, where a = 1.225 × 109 and b = 0.13. Calculate the total mass of the
Find the centroid of the given region assuming the density δ(x, y) = 1.Region bounded by y = 1 − x2 and y = 0
Find the centroid of the given region assuming the density δ(x, y) = 1.Region bounded by y2 = x + 4 and x = 4
Use a computer algebra system to compute numerically the centroid of the shaded region in Figure 12 bounded by r2=cos2θr2=cos⁡2θ for x≥0x≥0. 0.4 -0.4- - cos 20
Show that the centroid of the sector in Figure 13 has yyy-coordinate¯y=(2R3)(sinαα)y¯=(2R3)(sinαα)y¯=(2R3)(sin⁡αα) , (0.3) 8 R

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