Unit 11: Further Applications of Integration
Work, pressure, mass, and probability, once you notice they're all just another sum of infinitesimal slices.
5 topics · 22 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Work Done by a Variable Force
- How it worksHow does integrating force over distance add up all the tiny constant-force approximations into the true work?
- Common mistakeWhy doesn't the work done by a variable force equal the average force times the total distance as a simple average?
- Worked exampleFind the work done stretching a spring 0.5 meters beyond natural length with spring constant 200 N/m.
- Going furtherWhat happens to the work integral when a spring is compressed instead of stretched?
Your turn1 of 2
A spring has spring constant 309 N/m. Write the work W(x) done stretching it x meters beyond its natural length, as a function of x.Pumping Liquid Out of a Tank
- How it worksHow does slicing the tank into thin horizontal layers, each with its own lift distance, build the pumping integral?
- Going furtherWhat happens to the pumping-work integral when the liquid only needs to be raised partway instead of all the way out?
- Worked exampleFind the work required to pump all the water out of the top of a cylindrical tank of radius 2 m and height 5 m.
- Common mistakeWhy doesn't the work to pump out a tank depend only on the total weight of the liquid inside it?
Your turn1 of 2
A cylindrical tank of radius 3 m is full of water. Write the work integrand dW/dy (the work to lift a thin slice a distance y to the top), as a function of y.Hydrostatic Force on a Submerged Surface
- How it worksHow does integrating pressure times a strip's width and depth over the surface produce total hydrostatic force?
- Worked exampleFind the hydrostatic force on a vertical rectangular dam gate 3 m wide and 4 m tall, submerged from the surface.
- Going furtherWhat happens to the hydrostatic force integral when the submerged surface is triangular instead of rectangular?
- Common mistakeWhy isn't hydrostatic force just the water's weight pressing straight down on the surface?
Your turn1 of 2
A vertical rectangular dam gate is 5 m wide, submerged from the surface. Write the force integrand dF/dy (pressure times width at depth y), as a function of y.Center of Mass and Moments
- How it worksHow does dividing the total moment by the total mass locate exactly where a region's center of mass sits?
- Worked exampleFind the x-coordinate of the center of mass of the region bounded by y = x2 and y = 4, constant density.
- Common mistakeWhy doesn't the center of mass of an irregular region sit at the average of its leftmost and rightmost x-values?
- Worked exampleFind the center of mass of three point masses: 2 kg at x = 0, 3 kg at x = 4, and 5 kg at x = 6.
- Going furtherWhat happens to the center of mass calculation when the region has variable density instead of constant?
Your turn1 of 2
Three point masses lie on the x-axis: 4 kg at x=0, 4 kg at x=3, and a third mass 5 kg at position x. Write the center of mass as a function of x.Probability Density Functions
- How it worksHow does integrating a density function between two values give the probability of landing in that range?
- Worked exampleFind the probability that a random variable with density f(x) = 3x2 on [0, 1] falls between 0.2 and 0.5.
- Worked exampleFind the expected value of a random variable with density f(x) = 2x on the interval [0, 1].
- Common mistakeWhy doesn't the value of a probability density function at one exact point represent a probability?
- Going furtherWhat happens to a density function's integral when it's used to compute an expected value instead?
Your turn1 of 2
A random variable has the density f(x) = (n+1)xn on [0, 1], with n = 3. Write its cumulative distribution function F(x), the probability the variable is less than x.