A cable that weighs 6 lb/ft is used to lift 750 lb of coal up a mine shaft 500 ft deep. Find the work done.
Show how to approximate the required work by a Riemann sum.
To find the work done in lifting the coal, we need to calculate the integral of force with respect to distance over the entire length of the mine shaft.
Let's start by breaking down the problem into smaller intervals using a Riemann sum.
Step 1: Divide the mine shaft into smaller intervals.
In this case, we have a mine shaft that is 500 ft deep. We can divide it into smaller intervals of equal length. Let's say we choose a small interval width of Δx.
Step 2: Determine the height and weight in each interval.
The height of each interval will be Δx, and the weight in each interval can be approximated as 6 lb/ft * Δx.
Step 3: Calculate the work in each interval.
The work done in each interval is force times distance. In this case, the force is the weight in each interval, and the distance is the height of each interval. So the work in each interval is (6 lb/ft * Δx) * Δx = 6 lb * Δx^2.
Step 4: Sum up the work in each interval.
To approximate the total work done, we need to sum up the work in each interval. We can use a Riemann sum to do this. The Riemann sum is given by the formula:
Riemann sum = Σ(6 lb * Δx^2)
Step 5: Estimate the total work.
To find a more accurate estimate of the total work, we need to take the limit as Δx approaches zero. In other words, we need to find the definite integral of the function 6 lb * Δx^2 over the interval [0, 500].
Now, we can substitute the limits into the formula and evaluate the definite integral to find the exact work done.