A spring has a natural length of 2 m. It requires 100 J of work to stretch the spring to 7 m. Calculate the work required to stretch the spring from 6 m to 7 m. Round your answer to the nearest hundredth if necessary.
A cable with length 12 m has a mass of 24 kg and is hanging off of the side of a building. Acceleration due to gravity is g = 9.8 m/s^2. Find the work done in winding up the cable. Round your answer to the nearest tenth if necessary.
A force of 32 N is required to keep a spring stretched 2 m from the equilibrium position. How much work in Joules is done to stretch the spring 6 m from equilibrium? Round your answer to the nearest hundredth if necessary.
A crate has a mass of 5 kg. Using the fact that the acceleration due to gravity is g = 9.8 m/s^2, how much work is done in lifting the crate 4 m, in Joules? Round answer to one decimal place if necessary.
A cable with length 17 m has a mass of 34 kg and is hanging off of the side of a building. Acceleration due to gravity is g = 9.8 m/s^2. Find the work done in winding up the cable. Round your answer to the nearest tenth if necessary.
To lift a suitcase with a mass of 26 kg to the top of a building 2 m in height, a cable with length 2 m and a mass of 16 kg is used. An additional 2 m, with the same density, is used to secure the suitcase. Acceleration due to gravity is g = 9.8 m/s^2. How much work is done in lifting the suitcase to the top of the building? Round your answer to the nearest tenth if necessary.
A spring has a natural length of 1 m. It requires 180 J of work to stretch the spring to 7 m. Calculate the work required to stretch the spring from 6 m to 7 m. Round your answer to the nearest hundredth if necessary.
A spring has a natural length of 4 m. It requires 180 J of work to stretch the spring to 10 m. Calculate the work required to stretch the spring from 7 m to 11 m. Round your answer to the nearest hundredth if necessary.
A large water trough is 17 m long and has ends shaped like inverted isosceles triangles, with a base of 13 m and height of 10 m. Water density is 1000 kg/m^3 and acceleration due to gravity is 9.8 m/s^2. Find the force on one end of the tank if the trough is filled to a point 2 m below the edge. If necessary, round your answer to the nearest Newton.
The zoo is building a new polar bear exhibit and wants to put a semi-circular window in the concrete wall of the swimming tank. The window will be oriented so that the flat- edge of the semi-circle is the bottom of the window and parallel to the floor. If the semi-circle has a diameter of 14 m and the bottom of the window is at a depth of 21 m, find the hydrostatic force on the window. Water density is 1000 kg/m^3 and acceleration due to gravity is 9.8 m/s^2. If necessary, round your answer to the nearest Newton.
For a given cylindrical tank, the radius is 7 m and the height is 10 m. The tank is filled to a depth of 8 m. How much work is required to pump all of the water over the top edge of the tank? Acceleration due to gravity is 9.8 m/sec^2 and the density of water is 1000 kg/m^3. Round your answer to the nearest kilojoule.
For a given cylindrical tank, the radius is 2 m and the height is 7 m. The tank is filled to a depth of 6 m. How much work is required to pump all of the water over the top edge of the tank? Acceleration due to gravity is 9.8 m/sec^2 and the density of water is 1000 kg/m^3. Round your answer to the nearest kilojoule.
A large water trough is 17 m long and has ends shaped like inverted isosceles triangles, with a base of 14 m and height of 12 m. Water density is 1000 kg/m^3 and acceleration due to gravity is 9.8 m/s^2. Find the force on one end of the tank if the trough is completely full of water. If necessary, round your answer to the nearest Newton.
A large water trough is 19 m long and has ends shaped like inverted isosceles triangles, with a base of 12 m and height of 5 m. Water density is 1000 kg/m^3 and acceleration due to gravity is 9.8 m/s^2. Find the force on one end of the tank if the trough is filled to a point 1 m below the edge. If necessary, round your answer to the nearest Newton.
Find the force, in Newtons, on a rectangular metal plate with dimensions of 9 m by 10 m that is submerged horizontally in 17 m of water. Water density is 1000 kg/m3 and acceleration due to gravity is 9.8 m/s^2. If necessary, round your answer to the nearest Newton.
Find the force, in Newtons, on a rectangular metal plate with dimensions of 5 m by 9 m that is submerged horizontally in 17 m of water. Water density is 1000 kg/m^3 and acceleration due to gravity is 9.8 m/s^2. If necessary, round your answer to the nearest Newton.
For a given cylindrical tank, the radius is 2 m and the height is 13 m. The tank is filled to a depth of 8 m. How much work is required to pump all of the water over the top edge of the tank? Acceleration due to gravity is 9.8 m/sec^2 and the density of water is 1000 kg/m^3. Round your answer to the nearest kilojoule.
For a given cylindrical tank, the radius is 8 m and the height is 18 m. The tank is filled to a depth of 10 m. How much work is required to pump all of the water over the top edge of the tank? Acceleration due to gravity is 9.8 m/sec^2 and the density of water is 1000 kg/m^3. Round your answer to the nearest kilojoule.
For a given cylindrical tank, the radius is 4 m and the height is 11 m. The tank is filled to a depth of 7 m. How much work is required to pump all of the water over the top edge of the tank? Acceleration due to gravity is 9.8 m/sec^2 and the density of water is 1000 kg/m^3. Round your answer to the nearest kilojoule.
Find the moment M of the system with respect to the origin for the collection of masses given below. m1 = 14 kg at x1 = -2 m m2 = 24 kg at x2 = -17 m m3 = 35 kg at x3 = 10 m m4 = 10 kg at x4 = -19 m
Find the center of mass (x‾, y‾) of the system with respect to the origin for the collection of masses given below (assume the coordinates are given in meters). m1 = 7 kg at (1, -7) m2 = 2 kg at (7, -1) m3 = 12 kg at (-4, -5) Enter an exact answer using fractions when necessary.
Find the moments My and Mx of the system with respect to the origin for the collection of masses given below (assume the coordinates are given in meters). m1 = 14 kg at (1, -8) m2 = 5 kg at (-3, -6)
Calculate the center of mass for the collection of masses given below. m1 = 49 kg at x1 = -3 m m2 = 43 kg at x2 = -11 m m3 = 39 kg at x3 = -10 m m4 = 9 kg at x4 = 3 m Enter exact answer using fractions when necessary.
Find the center of mass (x‾, y‾) of the system with respect to the origin for the collection of masses given below (assume the coordinates are given in meters). m1 = 6 kg at (2, 6) m2 = 17 kg at (-5, -1) Enter an exact answer using fractions when necessary.
Find the moments My and Mx of the system with respect to the origin for the collection of masses given below (assume the coordinates are given in meters). m1 = 9 kg at (1, -8) m2 = 15 kg at (-7, -6)
Find the center of mass (x‾, y‾) of the system with respect to the origin for the collection of masses given below (assume the coordinates are given in meters). m1 = 14 kg at (8, 2) m2 = 17 kg at (5, -8) Enter an exact answer using fractions when necessary.
Find the moments My and Mx of the system with respect to the origin for the collection of masses given below (assume the coordinates are given in meters). m1 = 18 kg at (-3, -1) m2 = 14 kg at (-2, 7)
Calculate the center of mass for the collection of masses given below. m1 = 35 kg at x1 = 14 m m2 = 34 kg at x2 = 9 m m3 = 44 kg at x3 = 4 m m4 = 14 kg at x4 = -14 m Enter exact answer using fractions when necessary.
Find the moment M of the system with respect to the origin for the collection of masses given below. m1 = 17 kg at x1 = 20 m m2 = 19 kg at x2 = -20 m
Let R be the region bounded below by the graph of the function f(x) = 3x^2/4 - 3 and above by the x-axis over the interval [-2, 2]. Find the centroid, (x‾, y‾), of the region. Enter answer using exact values.
Let R be the region bounded below by the graph of the function f(x) = x^2/2 + 2 and above by the x-axis over the interval [-3, 3]. Find the centroid, (x‾, y‾), of the region. Enter answer using exact values.
Let R be the region bounded by the x-axis and the graph of the function f(x) = x^2 + 3 on the interval [2, 4]. Given uniform density ρ = 8, calculate the moments Mx and My of the region. Enter answer using exact value.
Let R be the region bounded above by the graph of the function f(x) = 7x + 2 and below by the x-axis over the interval [1, 6]. Find the total mass of the lamina given it has uniform density ρ = 3. Enter answer using exact value.
The region with vertices (6, 6), (12, 0), (20, 8), and (14, 14) is revolved about the x-axis to generate a solid. Find the volume of the solid. Enter answer using exact value.
Find the center of mass of a plate bounded above by the function f(x) = √576 – x^2 and below by the x-axis. Enter answer using exact values.
Find the centroid of the region bounded above by f(x) = x + 16 and below by g(x) = x^2 + 4. Enter answer using exact values.
Let R be a circle with the radius of 5 centered at (6, 0). Use the theorem of Pappus for volume to find the volume of the torus generated by revolving R around the y-axis.
Let R be the region bounded above by the graph of the function f(x) = 7x + 3 and below by the x-axis over the interval [2, 4]. Find the total mass of the lamina given it has uniform density ρ = 4. Enter answer using exact value.
Let R be the region bounded by the x-axis and the graph of the function f(x) = x^2 + 2 on the interval [2, 4]. Given uniform density ρ = 8, calculate the moments Mx and My of the region. Enter answer using exact value.
Let R be the region bounded below by the graph of the function f(x) = x^2/3 – 4/3 and above by the x-axis over the interval [-2, 2]. Find the centroid, (x‾, y‾), of the region. Enter answer using exact values.
Let R be the region bounded above by the graph of the function f(x) = -x^2 + 9 and below by the x-axis over the interval [-3, 3]. Find the centroid, (x‾, y‾), of the region. Enter answer using exact values.
The region with vertices (3, 3), (6, 0), (9, 3), and (6, 6) is revolved about the x-axis to generate a solid. Find the volume of the solid. Enter answer using exact value.
The region with vertices (4, 6), (10, 0), (22, 12), and (16, 18) is revolved about the x-axis to generate a solid. Find the volume of the solid. Enter answer using exact value.
Let R be a circle with the equation (x + 5)^2 + y^2 = 16. Use the theorem of Pappus for volume to find the volume of the torus generated by revolving R around the y-axis.
Find the centroid of the region bounded above by f(x) = 25 - x and below by g(x) = x^2 + 5. Enter answer using exact values.
Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius a = 3 is positioned with the left end of the circle at x = b = 7, and is rotated around the y-axis. Volume = Preview Write as an exact expression. Use pi for π.
Find the centroid of the region bounded above by f(x) = 9 – x^2 and below by g(x) = 3 - x. Enter answer using exact values.
Evaluate the integral below. ∫ 2x^2 e^4x dx Be sure to place the argument of any trigonometric or logarithmic functions in parentheses in your answer.
Evaluate the integral below. ∫ 7ln(8x^7 )/x^8 dx Be sure to place the argument of any trigonometric or logarithmic functions in parentheses in your answer.
Evaluate the integral below. ∫ tan^-1 (4x)dx Be sure to place the argument of any inverse trigonometric or logarithmic functions in parentheses in your answer.
Evaluate the integral below. ∫ 3x^2 e^2x dx Be sure to place the argument of any trigonometric or logarithmic functions in parentheses in your answer.
Evaluate the integral below using integration by parts given that u = 2x and dv = sin(2x)dx. ∫ 2xsin(2x)dx Be sure to place the argument of the trigonometric function in parentheses in your answer.
Evaluate the integral below using integration by parts given that u = ln(4x^4) and dv = 1/√x dx. ∫ ln(4x^4)/√x dx Be sure to place the argument of the logarithmic function in parentheses in your answer.
Evaluate the integral below. ∫ 5x^2 e^2xdx Be sure to place the argument of any trigonometric or logarithmic functions in parentheses in your answer.
Evaluate the integral below. ∫ 6x^2 e^4xdx Be sure to place the argument of any trigonometric or logarithmic functions in parentheses in your answer.
Evaluate the integral below using integration by parts given that u = 4x and dv = e^-5xdx. ∫ 4xe^-5xdx
Evaluate the integral below using integration by parts given that u = -x and dv = cos(2x)dx. ∫ -xcos(2x)dx Be sure to place the argument of the trigonometric function in parentheses in your answer.
Evaluate the integral below. ∫ 5x sec^2(x)dx Be sure to place the argument of any trigonometric or logarithmic functions in parentheses in your answer.
Evaluate the integral below. ∫ πsin^6(2πx) cos^3(2πx)dx Remember to put the arguments of the trigonometric functions in parentheses.
Evaluate the integral. ∫ 4tan^5(x) sec^3(x)dx Be sure to give the arguments of the trigonometric functions in parentheses.
Evaluate the integral below. ∫ -3πcos^3(πx) sin^3(πx)dx Be sure to put the argument of the trigonometric functions in parentheses.
Evaluate the integral. ∫ 2πtan^4(4πx) sec^6(4πx)dx Be sure to give the arguments of the trigonometric functions in parentheses
Which of the following substitutions should be made to evaluate the integral ∫ (4sin^2 (3x))(2cos^4 (3x))dx 4sin2 (3x) = 4 ( 1 2 - cos(6x) 2 ) and 2cos4 (3x) = 2 ( 1 2 + cos(6x) 2 ) 2 u = -cos(3x) 2cos4 (3x) = 2(1 - sin2 (3x)) 2 4sin2 (3x) = ( 1 2 - cos(3x) 2 ) and 2cos4 (3x) = - ( 1 2 + cos(3x) 2 ) 2
Evaluate the integral. ∫ - 5tan^5 (4x) sec^6 (4x)dx Be sure to give the arguments of the trigonometric functions in parentheses.
Evaluate the integral. ∫ - 2πtan^5 (πx) sec^3 (πx)dx Be sure to give the arguments of the trigonometric functions in parentheses.
Evaluate the integral. ∫ -2πtan^2 (πx) sec^6 (πx)dx Be sure to give the arguments of the trigonometric functions in parentheses.
Evaluate the following integral using a trigonometric substitution. ∫ 14 11 1/√36x^2 - 1296 dx Be sure to include absolute value signs around the arguments of logarithmic functions in your answer.
Evaluate the following integral. ∫ 1 0 1/√16x^2 + 576 dx Enter your answer as an exact answer. Be sure to include the argument of the trigonometric function in parentheses.
Evaluate the following integral using a trigonometric substitution. ∫ √1024 - 64x^2dx Be sure to include parentheses around the arguments of inverse trigonometric functions in your answer.
Evaluate the following integral using a trigonometric substitution. ∫ 1/√3136 - 49x^2 dx Be sure to include parentheses around the arguments of inverse trigonometric functions in your answer.
Evaluate the following integral using a trigonometric substitution. ∫ √4x^2 - 256dx Be sure to include absolute value signs around the arguments of any logarithmic functions in your answer.
Evaluate the following integral using a trigonometric substitution. ∫ √2304 - 36x^2dx Be sure to include parentheses around the arguments of inverse trigonometric functions in your answer.
Evaluate the following integral by completing the square and using a trigonometric substitution. ∫ x^2/√256 - 64x^2 dx Be sure to include parentheses around the arguments of inverse trigonometric functions in your answer.
Evaluate the following integral using a trigonometric substitution. ∫ 16 9 1/√81x^2 - 2025 dx Be sure to include absolute value signs around the arguments of logarithmic functions in your answer.
Evaluate the following integral using a trigonometric substitution. ∫ 9 5 1/√25x^2 - 225 dx Be sure to include absolute value signs around the arguments of logarithmic functions in your answer.
Evaluate the following integral using a trigonometric substitution. ∫ √36x^2 - 144dx Be sure to include absolute value signs around the arguments of any logarithmic functions in your answer.
Find the area of the region between the graph of f(x) = √x^2 - 9 and the x-axis over the interval [5, 14]. Enter your answer as an exact answer. Be sure to include the argument of the logarithmic function in parentheses.
Find the area of the region between the graph of f(x) = √x^2 - 9 and the x-axis over the interval [6, 13]. Enter your answer as an exact answer. Be sure to include the argument of the logarithmic function in parentheses.
Find the area of the region between the graph of f(x) = √x^2 - 9 and the x-axis over the interval [4, 8]. Enter your answer as an exact answer. Be sure to include the argument of the logarithmic function in parentheses.
Find the area of the region between the graph of f(x) = √x^2 - 16 and the x-axis over the interval [6, 14]. Enter your answer as an exact answer. Be sure to include the argument of the logarithmic function in parentheses.
Use the method of partial fractions to evaluate the following integral. ∫ 4x + 1 / x^2 + 2x - 3 dx Be sure to place the argument of any logarithmic functions in absolute value signs in your answer.
Use the method of partial fractions to evaluate the following integral. ∫ 4 / x^3 - 512 dx Be sure to place the argument of any logarithmic functions in absolute value signs when submitting your answer.
Evaluate the following integral using both long division and partial fraction methods. ∫ x^2 + x + 1 / x^2 - 4 dx Be sure to place the argument of any logarithmic functions in absolute value signs in your answer.
Find the volume of the solid generated by revolving the region enclosed by the graph of the function f(x) = 3 / x^2+5x+6 and the x-axis over the interval [0, 3] about y-axis. Enter an exact answer and be sure to place parentheses around the arguments of any logarithmic and trigonometric functions.
Find the volume of the solid generated by revolving the region enclosed by the graph of the function f(x) = 4 / x^2+7x+10 and the x-axis over the interval [0, 5] about y-axis. Enter an exact answer and be sure to place parentheses around the arguments of any logarithmic and trigonometric functions.
Use the method of partial fractions to evaluate the following integral. ∫ 4 / x^2(5x + 2) dx Be sure to place the argument of any logarithmic functions in absolute value signs when submitting your answer.
Evaluate the following integral using both long division and partial fraction methods. ∫ x^2 + x - 8 / x^2 - 9 dx Be sure to place the argument of any logarithmic functions in absolute value signs in your answer.
Use the method of partial fractions to evaluate the following integral. ∫ 3x + 1 / x^2 + x - 12 dx Be sure to place the argument of any logarithmic functions in absolute value signs in your answer.
Find the volume of the solid generated by revolving the region enclosed by the graph of the function f(x) = 6 / x^2+9x+18 and the x-axis over the interval [0, 1] about y-axis. Enter an exact answer and be sure to place parentheses around the arguments of any logarithmic and trigonometric functions.
Find the volume of the solid generated by revolving the region enclosed by the graph of the function f(x) = 6 / x^2+7x+10 and the x-axis over the interval [0, 4] about y-axis. Enter an exact answer and be sure to place parentheses around the arguments of any logarithmic and trigonometric functions.