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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Find the average value of the function over the interval.ƒ(x) = |x2 − 1|, [0, 4]
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the y-axis over the given interval.x = √sin y, x = 0; 0 ≤ y ≤ π
Calculate the volume of a cylinder inclined at an angle θ = 30° with height 10 and base of radius 4 (Figure 25). 30° 4 10
Sketch the enclosed region and use the Shell Method to calculate the volume of rotation about the x-axis.x = y, y = 0, x = 1
Find the work W required to empty the tank in Figure 8 through the hole at the top if the tank is half full of water. Water exits here. 8 4 5
Find the average value of the function over the interval.ƒ(x) =0 √9 − x2, [0, 3] Use geometry to evaluate the integral.
Find the area of the region lying to the right of x = y2 + 4y − 22 and to the left of x = 3y + 8.
Sketch the enclosed region and use the Shell Method to calculate the volume of rotation about the x-axis. x =y+1, x= 3-y, y=0
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the y-axis over the given interval.x = y2, x = √y
The areas of cross sections of Lake Nogebow at 5-m intervals are given in the table below. Figure 26 shows a contour map of the lake. Estimate the volume V of the lake by taking the average of the
Find the area of the region lying to the right of x = y2 − 5 and to the left of x = 3 − y2.
Find the average value of the function over the interval.ƒ(x) = x[x] , [0, 3], where [x] is the greatest integer function
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the y-axis over the given interval.x = 4 − y, x = 16 − y2
Sketch the enclosed region and use the Shell Method to calculate the volume of rotation about the x-axis.y(4 − y), x = 0
Figure 21 shows the region enclosed by x = y3 − 26y + 10 and x = 40 − 6y2 − y3. Match the equations with the curves and compute the area of the region. 3 -1 y ·X
Find the total mass of a 1-m rod whose linear density function is ρ(x) = 10(x + 1)−2 kg/m for 0 ≤ x ≤ 1.
Assume the tank in Figure 10 is full. Find the work required to pump out half of the water. Hint: First, determine the level H at which the water remaining in the tank is equal to one-half the total
Find the average value of the function over the interval.Find ∫52 g(t) dt if the average value of g on [2, 5] is 9.
Rotation of the region in Figure 12 about the y-axis produces a solid with two types of different cross sections. Compute the volume as a sum of two integrals, one for −12 ≤ y ≤ 4 and one for 4
Assume that the tank in Figure 10 is full.(a) Calculate the work F(y) required to pump out water until the water level has reached level y.(b) Plot F.(c) What is the significance of F'(y) as a rate
Sketch the enclosed region and use the Shell Method to calculate the volume of rotation about the x-axis.x = y(4 − y), x = (y − 2)2
Figure 22 shows the region enclosed by y = x3 − 6x and y = 8 − 3x2.Match the equations with the curves and compute the area of the region. -3 y -50- 2 X
Find the total mass of a 3-m rod whose linear density function is ρ(x) = 3 + cos(πx) kg/m for 0 ≤ x ≤ 3.
The average value of R over [0, x] is equal to x for all x. Use the FTC to determine R(x).
Let R be the region enclosed by y = x2 + 2, y = (x − 2)2 and the axes x = 0 and y = 0. Compute the volume V obtained by rotating R about the x-axis. Express V as a sum of two integrals.
Calculate the work required to lift a 10-m chain over the side of a building (Figure 13). Assume that the chain has a density of 8 kg/m. Break up the chain into N segments, estimate the work
Sketch the enclosed region and use the Shell Method to calculate the volume of rotation about the x-axis.y = 4 − x2, x = 0, y = 0
A mineral deposit along a strip of length 6 cm has density s(x) = 0.01x(6 − x) g/cm for 0 ≤ x ≤ 6. Calculate the total mass of the deposit.
Use the Washer Method to find the volume obtained by rotating the region in Figure 3 about the x-axis. y y=x² y=mx x+
Find the area enclosed by the graphs in two ways: by integrating along the x-axis and by integrating along the y-axis.x = 9 − y2, x = 5
Find the volume of the solid obtained by rotating region A in Figure 13 about the given axis.x-axis 6 2- A 1 y=x2+2 B 2 X
Find the volume of the solid obtained by rotating region A in Figure 13 about the given axis.y = −2 6 2- A 1 y=x2+2 B 2 X
How much work is done lifting a 3-m chain over the side of a building if the chain has mass density 4 kg/m?
Sketch the enclosed region and use the Shell Method to calculate the volume of rotation about the x-axis.y = x1/3 − 2, y = 0, x = 27
Charge is distributed along a glass tube of length 10 cm with linear charge density ρ(x) = x(x2 + 1)−2 × 10−4 coulombs per centimeter (C/cm) for 0 ≤ x ≤ 10. Calculate the total charge.
Use the Shell Method to find the volume obtained by rotating the region in Figure 3 about the x-axis. y y=x² y=mx -X
The semicubical parabola y2 = x3 and the line x = 1
Find the volume of the solid obtained by rotating region A in Figure 13 about the given axis.y = 2 6 2- A 1 y=x2+2 B 2 X
Determine which of the following is the appropriate integrand needed to determine the volume of the solid obtained by rotating around the vertical axis given by x = −1 the area that is between the
A 6-m chain has mass 18 kg. Find the work required to lift the chain over the side of a building.
Calculate the population within a 10-mile radius of the city center if the radial population density is ρ(r) = 4(1 + r2)1/3 (in thousands per square mile).
Find the area of the region using the method (integration along either the x- or the y-axis) that requires you to evaluate just one integral.Region between y2 = x + 5 and y2 = 3 − x
Use any method to find the volume of the solid obtained by rotating the region enclosed by the curves about the given axis.y = x2 + 2, y = x + 4, x-axis
Find the volume of the solid obtained by rotating region A in Figure 13 about the given axis.y-axis 6 2- A 1 y=x2+2 B 2 X
A 10-m chain with mass density 4 kg/m is initially coiled on the ground. How much work is performed in lifting the chain so that it is fully extended (and one end touches the ground)?
Let y = ƒ(x) be a decreasing function on [0, b], such that ƒ(b) = 0. Explain why where h denotes the inverse of ƒ. 2π f xf(x) dx = π fƒ© (h(x))² dx,
Odzala National Park in the Republic of the Congo has a high density of gorillas. Suppose that the population density is given by the radial density function ρ(r) = 52(1 + r2)−2 gorillas/km2,
Find the area of the region using the method (integration along either the x- or the y-axis) that requires you to evaluate just one integral.Region between y = x and x + y = 8 over [2, 3]
Find the volume of the solid obtained by rotating region A in Figure 13 about the given axis.x = −3 6 2- A 1 y=x2+2 B 2 X
Use any method to find the volume of the solid obtained by rotating the region enclosed by the curves about the given axis.y = x2 + 6, y = 8x − 1, y-axis
How much work is done lifting a 12-m chain that has mass density 3 kg/m (initially coiled on the ground) so that its top end is 10 m above the ground?
Table 1 lists the population density (in people per square kilometer) as a function of distance r (in kilometers) from the center of a rural town. Estimate the total population within a 1.2-km radius
Use both the Shell and Disk Methods to calculate the volume obtained by rotating the region under the graph of ƒ(x) = 8 − x3 for 0 ≤ x ≤ 2 about(a) The x-axis (b) The y-axis
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.y = 25 − x2, y = x2 − 25
Use any method to find the volume of the solid obtained by rotating the region enclosed by the curves about the given axis.x = y2 − 3, x = 2y, axisy = 4
Find the volume of the solid obtained by rotating region A in Figure 13 about the given axis.x = 2 6 2- A 1 y=x2+2 B 2 X
Use any method to find the volume of the solid obtained by rotating the region enclosed by the curves about the given axis.y = 2x, y = 0, x = 8, axis x = −3
A 500-kg wrecking ball hangs from a 12-m cable of density 15 kg/m attached to a crane. Calculate the work done if the crane lifts the ball from ground level to 12 m in the air by drawing in the cable.
Sketch the solid of rotation about the y-axis for the region under the graph of the constant function ƒ(x) = c (where c > 0) for 0 ≤ x ≤ r.(a) Find the volume without using integration.(b) Use
Find the total mass of a circular plate of radius 20 cm whose mass density is the radial function ρ(r) = 0.03 + 0.01 cos(πr2) g/cm2.
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.y = x2 − 6, y = 6 − x3, x = 0
Find the volume of the solid obtained by rotating region B in Figure 13 about the given axis.x-axis 6 2 У A y=x2+2 B 2 - Х
Use any method to find the volume of the solid obtained by rotating the region enclosed by the curves about the given axis.y = x2 − 1, y = 2x − 1, axis x = −2
Calculate the work required to lift a 3-m chain over the side of a building if the chain has a variable density of ρ(x) = x2 − 3x + 10 kg/m for 0 ≤ x ≤ 3.
The density of deer in a forest is the radial function ρ(r) = 150(r2 + 2)−2 deer per square kilometer, where r is the distance (in kilometers) to a small meadow. Calculate the number of deer in
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.x + y = 4, x − y = 0, y + 3x = 4
Find the volume of the solid obtained by rotating region B in Figure 13 about the given axis.y = −2 6 2 A y=x² + 2 B 2 X
Use any method to find the volume of the solid obtained by rotating the region enclosed by the curves about the given axis.y = x2 − 1, y = 2x − 1, axis y = 4
A 3-m chain with linear mass density ρ(x) = 2x(4 − x) kg/m lies on the ground. Calculate the work required to lift the chain from its front end so that its bottom is 2 m above ground.
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.y = 8 − 3x, y = 6 − x, y = 2
Show that a circular plate of radius 2 cm with radial mass density ρ(r) = 4 r g/cm2 has finite total mass, even though the density becomes infinite at the origin.
Let W be the volume of the solid obtained by rotating the region under the graph in Figure 11(B) about the y-axis.(a) Describe the figures generated by rotating segments A'B' and A'C' about the
Find the volume of the solid obtained by rotating region B in Figure 13 about the given axis.y = 6 6 2 A y=x² + 2 B 2 X
Use any method to find the volume of the solid obtained by rotating the region enclosed by the curves about the given axis.y = −x2 + 4x − 3, y = 0, axis y = −1
The gravitational force between two objects of mass m and M, separated by a distance r, has magnitude GMm/r2, where G = 6.67 × 10−11 m3kg−1s−1.Show that if two objects of mass M and m are
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.y = 15 −√x, y = 2 √ x, x = 0
Find the flow rate through a tube of radius 4 cm, assuming that the velocity of fluid particles at a distance r centimeters from the center is v(r) = (16 − r2) cm/s.
Let R be the region under the graph of y = 9 − x2 for 0 ≤ x ≤ 2. Use the Shell Method to compute the volume of rotation of R about the x-axis as a sum of two integrals along the y-axis. The
Find the volume of the solid obtained by rotating region B in Figure 13 about the given axis.y-axis 6 2 A y=x² + 2 B 2 X
Use any method to find the volume of the solid obtained by rotating the region enclosed by the curves about the given axis.y = −x2 + 4x − 3, y = 0, axis x = 4
Use the result of Exercise 35 to calculate the work required to place a 2000-kg satellite in an orbit 1200 km above the surface of the earth. Assume that the earth is a sphere of radius Re = 6.37 ×
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.y = |x2 − 4|, y = 5
The velocity of fluid particles flowing through a tube of radius 5 cm is v(r) = (10 − 0.3r − 0.34r2) cm/s, where r centimeters is the distance from the center. What quantity per second of fluid
Let R be the region under the graph of y = 4x−1 for 1 ≤ y ≤ 4. Use the Shell Method to compute the volume of rotation of R about the y-axis as a sum of two integrals along the x-axis.
Find the volume of the solid obtained by rotating region B in Figure 13 about the given axis.x = 2 6 2 A y=x² + 2 B 2 X
Let v(r) be the velocity of blood in an arterial capillary of radius R = 4 × 10−5 m. Use Poiseuille’s Law (Example 6) with k = 106 (m-s)−1 to determine the velocity at the center of the
Use any method to find the volume of the solid obtained by rotating the region enclosed by the curves about the given axis.x = 4y − y3, x = 0, y ≥ 0, x-axis
Use the result of Exercise 35 to compute the work required to move a 1500-kg satellite from an orbit 1000 to an orbit 1500 km above the surface of the earth.Data From Exercise 35The gravitational
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.x = |y|, x = 1 − |y|
A solid rod of radius 1 cm is placed in a pipe of radius 3 cm so that their axes are aligned. Water flows through the pipe and around the rod. Find the flow rate if the velocity of the water is given
Use the Shell Method to find the volume obtained by rotating region A in Figure 12 about the given axis.y-axis 6 2 У A 1 B y=x2+2 2 X
Find the volume of the solid obtained by rotating region B in Figure 13 about the given axis.x = −3 6 2 A y=x² + 2 B 2 X
Use any method to find the volume of the solid obtained by rotating the region enclosed by the curves about the given axis.y2 = x−1, x = 1, x = 3, axis y = −3
Use the Shell Method to find the volume obtained by rotating region A in Figure 12 about the given axis.x = −3 6 2 У A 1 B y=x2+2 2 X
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.y = |x|, y = x2 + 3
The pressure P and volume V of the gas in a cylinder of length 0.8 m and radius 0.2 m, with a movable piston, are related by PV1.4 = k, where k is a constant (Figure 14). When the piston is fully
Use any method to find the volume of the solid obtained by rotating the region enclosed by the curves about the given axis.y = cos(x2), y = 0, 0 ≤ x ≤ √π/2, y-axis
Use the Shell Method to find the volume obtained by rotating region A in Figure 12 about the given axis.x = 2 6 2 У A 1 B y=x2+2 2 X
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.y = x2, y = 12 − x, x = 0, about y = −2 x ≥ 0
Sketch the region enclosed by the curves and compute its area as an integral along the x-or y-axis.x = y3 − 18y, y + 2x = 0

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