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.

**Related Pages**

Integral Calculus

Calculus: Integration

Calculus: Derivatives

Calculus Lessons

**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/m^{3}. The height of the oil is 10 m. How much work is involved in pumping all the oil out the top of the tank?

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