Math Archive: Questions from 2024-01-4

Consider a thin rod oriented on the x-axis over the interval [-4, 3], where x is in meters. If the density of the rod is given by the function ρ(x) = x^2 + 1, in kilograms per meter, what is the mass of the rod in kilograms? Enter your answer as an exact value.

Consider a thin rod oriented on the x-axis over the interval [0, 1], where x is in meters. If the density of the rod is given by the function ρ(x) = 7x^3/4 + 8, in kilograms per meter, what is the mass of the rod in kilograms? Enter your answer as an exact value.

Consider a plate with a radius of 8 and a radial density given by ρ(x) = 4x^2/3 + 5. What is the mass of the plate? (Enter answer using exact value.)

Consider a plate with a radius of 10 and a radial density given by ρ(x) = 18cos(x^2). What is the mass of the plate? (Enter answer using exact value.)

Consider a thin rod oriented on the x-axis over the interval [2π/3, 5π/3], where x is in meters. If the density of the rod is given by the function ρ(x) = 4sin(x) + 5, in kilograms per meter, what is the mass of the rod in kilograms? Enter your answer as an exact value.

Consider a thin rod oriented on the x-axis over the interval [1, 5], where x is in meters. If the density of the rod is given by the function ρ(x) = 3 + 4x^-4 , in kilograms per meter, what is the mass of the rod in kilograms? Enter your answer as an exact value.

Consider a thin rod oriented on the x-axis over the interval [2, 6], where x is in meters. If the density of the rod is given by the function ρ(x) = 6 + e^3x , in kilograms per meter, what is the mass of the rod in kilograms? Enter your answer as an exact value.

Consider a plate with a radius of 9 and a radial density given by ρ(x) = 5x^1/2 + 7. What is the mass of the plate? (Enter answer using exact value.)

Consider a plate with a radius of 8 and a radial density given by ρ(x) = 2x^2/3 + 4. What is the mass of the plate? (Enter answer using exact value.)

Consider a plate with a radius of 3 and a radial density given by ρ(x) = 4x + 2. What is the mass of the plate? (Enter answer using exact value.)

Consider a plate with a radius of 8 and a radial density given by ρ(x) = 6 + 2x^-4/3 . What is the mass of the plate? (Enter answer using exact value.)

Consider a plate with a radius of 2 and a radial density given by ρ(x) = 6 + e 2x^2 . What is the mass of the plate? (Enter answer using exact value.)

To lift a basket with a mass of 44 kg to the top of a building 2 m in height, a cable with length 2 m and a mass of 12 kg is used. Acceleration due to gravity is g = 9.8 m/s^2. How much work is done in lifting the basket to the top of the building? Round your answer to the nearest tenth if necessary.

The force in Newtons acting on a particle located x meters from the origin is given by the function F(x) = 21x 2 + 3. What is the work done in Joules when moving the particle from x = 5 to x = 7?

A suitcase has a mass of 4 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 suitcase 2 m, in Joules? Round answer to one decimal place if necessary.