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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Find the centroid of the given solid region assuming a density of δ(x,y)=1δ(x,y)=1.Hemisphere x2 + y2 + z2 ≤ R2, z ≥ 0
Find the centroid of the given solid region assuming a density of δ(x,y)=1δ(x,y)=1δ(x,y)=1.Region bounded by the xyxyxy-plane, the cylinder x2+y2=R2x2+y2=R2x2+y2=R2, and the plane
Find the centroid of the given solid region assuming a density of δ(x,y)=1δ(x,y)=1.The “ice cream cone” region W bounded, in spherical coordinates, by the cone ϕ = π/3 and the sphere ρ = 2
Use the result of Exercise 45 to calculate the radius of gyration of a uniform sphere of radius R about any axis through the center of the sphere.Data From Exercise 45 Show that the moment of
Prove the formula for the right circular cylinder in Figure 18.Iz=12MR2Iz=12MR2 R H
Prove the formula for the right circular cylinder in Figure 18.\(I_{x}=\frac{1}{4} M R^{2}+\frac{1}{12} M H^{2}\) R H
The yo-yo in Figure 19 is made up of two disks of radius r = 3 cm and an axle of radius b = 1 cm. Each disk has mass M1 = 20 g, and the axle has mass M2 = 5 g.(a) Use the result of Exercise 47 to
Calculate Iz for the solid region W inside the hyperboloid x2 + y2 = z2 + 1 between z = 0 and z = 1.
Sketch the region D indicated and integrate ƒ(x, y) over D using polar coordinates. f(x,y)= x + y, x + y 2
Sketch the region D indicated and integrate ƒ(x, y) over D using polar coordinates. f(x, y) = x + y; 1 x + y 4
Sketch the region D indicated and integrate ƒ(x, y) over D using polar coordinates. f(x,y) = xy; x 0, y 0, x + y 4
Sketch the region D indicated and integrate ƒ(x, y) over D using polar coordinates. f(x, y) = y(x + y); y 0, x + y 1
Sketch the region D indicated and integrate ƒ(x, y) over D using polar coordinates. f(x,y) = y(x + y)-; y , x + y 1
Sketch the region D indicated and integrate ƒ(x, y) over D using polar coordinates. f(x, y) = x+x; x + y R
Sketch the region of integration and evaluate by changing to polar coordinates. J-2 4-1 (x + y) dy dx
Sketch the region of integration and evaluate by changing to polar coordinates. Jo Jo 9-y2 x +ydxdy
Sketch the region of integration and evaluate by changing to polar coordinates. Jo 1/2 -1-x 3x x dydx
Sketch the region of integration and evaluate by changing to polar coordinates. 16-x S S tan y - dy dx X
Sketch the region of integration and evaluate by changing to polar coordinates. S S fxdx dy
Sketch the region of integration and evaluate by changing to polar coordinates. 'x v3x ydy dx
Sketch the region of integration and evaluate by changing to polar coordinates. S.S. 0 4-x (x + y) dy dx
Calculate the integral over the given region by changing to polar coordinates. (x, y) = (x + y)-; x + y 2, x1
Calculate the integral over the given region by changing to polar coordinates. f(x,y) = y; 2 x + y 9
Calculate the integral over the given region by changing to polar coordinates. f(x, y) = xyl; x + y 1
Calculate the integral over the given region by changing to polar coordinates. f(x, y) = (x + y)-3/; x + y 1, x+yl
Calculate the integral over the given region by changing to polar coordinates. f(x,y) = x-y; x + y 1, x+y 1
Calculate the integral over the given region by changing to polar coordinates. f(x,y) = y; x + y 1, (x-1) + y 1
Find the volume of the wedge-shaped region (Figure 18) contained in the cylinder x2 + y2 = 9, bounded above by the plane z = x and below by the xy-plane. X X=Z
Let W be the region above the sphere x2 + y2 + z2 = 6 and below the paraboloid z = 4 − x2 − y2.(a) Show that the projection of W on the xy-plane is the disk x2 + y2 ≤ 2 (Figure 19).(b)
Evaluate where D is the domain in Figure 20. Find the equation of the inner circle in polar coordinates and treat the right and left parts of the region separately. SS x + y dA,
Evaluate where D is the shaded region enclosed by the lemniscate curve r2 = sin 2θ in Figure 21. SS xx. x + ydA, X
Let W be the region above the plane z = 2 and below the paraboloid z = 6 − (x2 + y2).(a) Describe W in cylindrical coordinates.(b) Use cylindrical coordinates to compute the volume of W.
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 4z, z x + y
Use spherical coordinates to calculate the triple integral of ƒ(x, y, z) over the given region. f(x,y,z) = x + y + z; x + y + z 2z
Use spherical coordinates to calculate the triple integral of ƒ(x, y, z) over the given region. f(x, y, z)= p; x + y +2 4, z1, x20
Use spherical coordinates to evaluate the triple integral of ƒ(x, y, z) = z over the region 03057353 05057 2' 1p2
What is the mass density δ(x, y, z) of a solid of volume 5 m3 with uniform mass density and total mass 25 kg?
A domain D in R2 with uniform mass density is symmetric with respect to the y-axis. Which of the following are true?(a) xCM = 0 (b) yCM = 0 (c) Ix = 0 (d) Iy = 0
If p(x, y) is the joint probability density function of random variables X and Y, what does the double integral of p(x, y) over [0, 1] × [0, 1] represent? What does the integral of p(x, y) over the
Compute the Riemann sum S4,3 to estimate the double integral of ƒ(x, y) = xy over R = [1, 3] × [1, 2.5]. Use the regular partition and upper-right vertices of the subrectangles as sample points.
Compute the Riemann sum with N = M = 2 to estimate the integral of √x + y over R = [0, 1] × [0, 1]. Use the regular partition and midpoints of the subrectangles as sample points.
Compute the Riemann sums for the double integral for the grid and two choices of sample points shown in Figure 16.ƒ(x, y) = 2x + y SS f(x,y) dA, where R = [1,4] [1,3], R
Compute the Riemann sums for the double integral for the grid and two choices of sample points shown in Figure 16.ƒ(x, y) = 7 SS f(x,y) dA, where R = [1,4] [1,3], R
Compute the Riemann sums for the double integral for the grid and two choices of sample points shown in Figure 16.ƒ(x, y) = 4x SS f(x,y) dA, where R = [1,4] [1,3], R
Compute the Riemann sums for the double integral for the grid and two choices of sample points shown in Figure 16.ƒ(x, y) = x − 2y SS f(x,y) dA, where R = [1,4] [1,3], R
Let R = [0, 1] × [0, 1]. Estimate by computing two different Riemann sums, each with at least six rectangles. JR (x + y) dA
Evaluate where R = [2, 5] × [4, 7]. R 4dA,
Evaluate where R = [0, 5] × [0, 3], and sketch the corresponding solid region (see Example 2). JR (15 - 3x)dA,
Evaluate where R = [2, 5] × [4, 7]. SS (-5) dA, R
The following table gives the approximate height at quarter-meter intervals of a mound of gravel. Estimate the volume of the mound by computing the average of the two Riemann sums S 4,3 with
Use the following table to compute a Riemann sum S3,3 for ƒ(x, y) on the square R = [0, 1.5] × [0.5, 2]. Use the regular partition and sample points of your choosing. 2 1.5 1 0.5 0 y x Values of f
Let SN,N be the Riemann sum fordy dx using the regular partition and the lower left vertex of each subrectangle as sample points. Use a computer algebra system to calculate SN,N for N = 25, 50, 100.
Let SN,M be the Riemann sum for using the regular partition and the upper-right vertex of each subrectangle as sample points. Use a computeralgebra system to calculate S2N,N for N = 25, 50, 100. 0
Evaluate the integral. SS=dA. dA, R= [-2, 4] x [1,3] JRY
Evaluate the integral. R xydA, R= [1,1] [0,2]
Evaluate the integral. cos x sin 2y dA, R= [0, ] [0,]
Evaluate the integral. SS dA, R= [0,2] [0,4] y x+1
Evaluate the integral. SS et siny dA, R= [0,2] [0, 1] R
Evaluate the integral. SS ex+4y dA, R= [0, 1] [1,2]
Evaluate the integral. SS xlny dA, R= [0, 3] [1, e]
Evaluate the integral. IR xtany dA, R= [0,2] [0, 1]
Let ƒ(x, y) = mxy2, where m is a constant. Find a value of m such that where R = [0, 1] × [0, 2]. SS f(x,y)dA = 1, JR
Evaluate You will need Integration by Parts and the formulaThen evaluate I again using Fubini’s Theorem to change the order of integration (i.e., integrate first with respect to x). Which method is
(a) Which is easier, antidifferentiating y√1 + xy with respect to x or with respect to y? Explain.(b) Evaluate y1 + xydA, where R= [0, 1] [0, 1].
(a) Which is easier, antidifferentiating xexy with respect to x or with respect to y? Explain.(b) Evaluate SS xey dA, where R= [0, 1] x [0, 1]. 'R
(a) Which is easier, antidifferentiating y/1+xy with respect to x or with respect to y? Explain.(b) Evaluate y SS17 1 + xy dA, where R = [0, 1] [0, 1].
Calculate a Riemann sum S3,3 on the square R = [0, 3] × [0, 3] for the function ƒ(x, y) whose contour plot is shown in Figure 17. Choose sample points and use the plot to find the values of ƒ(x,
Using Fubini’s Theorem, argue that the solid in Figure 18 has volume AL, where A is the area of the front face of the solid. THEOREM 3 Fubini's Theorem The double integral of a continuous function
Prove the following extension of the Fundamental Theorem of Calculus to two variables: If ∂2F/∂x ∂y = ƒ(x, y), thenwhere R = [a, b] × [c, d]. R f(x,y) dA= F(b,d) - F(a,d) - F(b,c) + F(a, c)
Let F(x, y) = x−1exy. Show that ∂2F/∂x ∂y = yexy and use the result of Exercise 52 to evaluate for R = [1, 3] × [0, 1].Data From Exercise 52Prove the following extension of the Fundamental
Find a function F(x, y) satisfying and use the result of Exercise 52 to evaluate for R = [0, 1] × [0, 4].Data From Exercise 52Prove the following extension of the Fundamental Theorem of Calculus
In this exercise, we use double integration to evaluate the following improper integral for a > 0 a positive constant: I(a) = (c) Show that I (a): So exe-ax exy dydx. -*- - e X (a) Use L'Hpital's
Which of the following expressions do not make sense? (a) (c) S S f(x,y) dy dx So f(x,y) dydx (b) (d) f(x,y) dy dx S. So SS f(x. y) dy dx
Draw a domain in the plane that is neither vertically nor horizontally simple.
Which of the four regions in Figure 21 is the domain of integration -1- LY -x -2/2. f(x,y) dy dx?
Let D be the unit disk. If the maximum value of ƒ(x, y) on D is 4, then the largest possible value of is (choose the correct answer):(a) 4 (b) 4π (c) 4/π D f(x,y) dA
Calculate the Riemann sum for ƒ(x, y) = x − y and the shaded domain D in Figure 22 with two choices of sample points, • and ◦. Which do you think is a better approximation to the integral of
Approximate values of f (x, y) at sample points on a grid are given in Figure 23. Estimate for the shaded domain by computing the Riemann sum with the given sample points. JJD f(x, y) dx dy
Express the domain D in Figure 24 as both a vertically simple region and a horizontally simple region, and evaluate the integral of ƒ(x, y) = xy over D as an iterated integral in two ways. 1 y y = 1
Sketch the domainand evaluate as an iterated integral. D: 0x1, x y 4-x
Compute the double integral of ƒ(x, y) = x2y over the given shaded domain in Figure 25.(A) 2 e 2 3 4 (A) -X 21 2 + 1 + 2 3 4 (B) -X 21 2 + + 12 34 (C) -X
Compute the double integral of ƒ(x, y) = x2y over the given shaded domain in Figure 25.(B) 2 e 2 3 4 (A) -X 21 2 + 1 + 2 3 4 (B) -X 21 2 + + 12 34 (C) -X
Compute the double integral of ƒ(x, y) = x2y over the given shaded domain in Figure 25.(C) 2 e 2 3 4 (A) -X 21 2 + 1 + 2 3 4 (B) -X 21 2 + + 12 34 (C) -X
Sketch the domain D defined by x + y ≤ 12, x ≥ 4, y ≥ 4 and compute exty dA.
Integrate ƒ(x, y) = x over the region bounded by y = x2 and y = x + 2.
Sketch the region D between y = x2 and y = x(1 − x). Express D as a simple region and calculate the integral of ƒ(x, y) = 2y over D.
Evaluate where D is the shaded part inside the semicircle of radius 2 in Figure 26. D y - dA, X
Calculate the double integral of ƒ(x, y) = y2 over the rhombus R in Figure 27. -2 y -4+ R 2 -X
Calculate the double integral of ƒ(x, y) = x + y over the domain D = {(x, y) : x2 + y2 ≤ 4, y ≥ 0}.
Integrate ƒ(x, y) = (x + y + 1)−2 over the triangle with vertices (0, 0), (4, 0), and (0, 8).
Calculate the integral of ƒ(x, y) = x over the region D bounded above by y = x(2 − x) and below by x = y(2 − y). Apply the quadratic formula to the lower boundary curve to solve for y as a
Integrate ƒ(x, y) = x over the region bounded by y = x, y = 4x − x2, and y = 0 in two ways: as a vertically asimple region and as a horizontally simple region.
Compute the double integral of ƒ(x, y) over the domain D indicated. f(x,y) = xy; 0x 5, x y 2x + 3
Compute the double integral of ƒ(x, y) over the domain D indicated. f(x, y)= -2; 0 x 3, 1yet
Compute the double integral of ƒ(x, y) over the domain D indicated. f(x,y) = x; 0x 1, 1 y et
Compute the double integral of ƒ(x, y) over the domain D indicated. f(x, y) = cos(2x+y); x, 1 y 2x
Compute the double integral of ƒ(x, y) over the domain D indicated. f(x, y) = 6xy - x; bounded below by y = x, above by y = x
Compute the double integral of ƒ(x, y) over the domain D indicated.ƒ(x, y) = sin x; bounded by x = 0, x = 1, y = 0, y = cos x
Compute the double integral of ƒ(x, y) over the domain D indicated.ƒ(x, y) = ex+y; bounded by y = x − 1, y = 12 − x for 2 ≤ y ≤ 4
Compute the double integral of ƒ(x, y) over the domain D indicated.ƒ(x, y) = (x + y)−1; bounded by y = x, y = 1, y = e, x = 0

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