10. Physics Applications of Integrals

How much work is required to push a chair across the floor with a force of $F=8\mathrm{lb}$ from $x=6\mathrm{ft}$ to $x=9\mathrm{ft}$ along the $x$-axis?

How much work is done by a person lifting a $10\mathrm{lb}$ bucket $4\mathrm{ft}$ off the ground?

How much work is required to move an object with a force of $F\left(x\right)=4x^{3}N$ acting along the $x$-axis from $x=0m$ to $x=2m$?

Compute the work done by a force of $F=\frac{3}{x^2}N$ from $x=2m$ to $x=6m$.

90. Work Let R be the region in the first quadrant bounded by the curve y = √(x⁴ - 4)

and the lines y = 0 and y = 2. Suppose a tank that is full of water has the shape of a solid of revolution obtained by revolving region R about the y-axis. How much work is required to pump all the water to the top of the tank? Assume x and y are in meters.

Explain how to find the mass of a one-dimensional object with a variable density ρ.

Why is integration used to find the work required to pump water out of a tank?

9–12. Consider the cylindrical tank in Example 4 that has a height of 10 m and a radius of 5 m. Recall that if the tank is full of water, then ∫₀¹⁰ 25 π ρg(15−y) dy equals the work required to pump all the water out of the tank, through an outflow pipe that is 15 m above the bottom of the tank. Revise this work integral for the following scenarios. (Do not evaluate the integrals.)

The work required to empty the top half of the tank