# Work Done using Calculus - Tank Problems

A series of free Calculus Video Lessons:

• How to Calculate the Work Required to Drain a Tank Using Calculus?
• How to Using integration to calculate the amount of work done pumping fluid?
• How to find the work required to lift a rope to the top of a building.

How to calculate the work done in stretching a spring using Hooke’s Law and a definite integral?
The force required to maintain a spring stretched x units beyond its natural length is proportional to x (let k be the constant of proportionality) so we get F = kx.

Example:
A spring has a natural length of 20 cm. If a 25 Newton force is required to keep it stretched to 30 cm, how much work is required to stretch it from 20 cm to 25 cm.

How to use a definite integral to calculate the work done in raising a leaky bucket 20 feet?

Example:
A leaky 5 pound bucket is lifted 20 feet into the air at a constant speed. The rope weighs 0.3 lbs/ft. The bucket starts with 10 pounds of water and leaks at a constant rate. There are 5 pounds of water left as it reaches the top. How much work was done to raise the bucket?

Calculating the Work Required to Drain a Tank - Using Calculus
One complete example is shown along with a general procedure to follow.

Applications of Integrals - Pumping (Work)
Using integration to calculate the amount of work done pumping fluid.

Lifting a Leaky Bag of Sand
Demonstrates the use of integration to calculate the work done lifting a leaky bag of sand.

Finding Work using Calculus - The Cable/Rope Problem
This video shows how to find the work required to lift a rope to the top of a building.

Finding Work using Calculus - The Cable/Rope Problem - Part b
In this video, I find the work required to lift up only HALF of the rope to the top of the building.

Pump oil from inverted cone
Example:
An inverted conical tank with a height of 20 m and a base diameter of 25 m contains oil with density 800 kg/m3. The height of the oil is 10 m. How much work is involved in pumping all the oil out the top of the tank?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 