What is the difference between "area" and "net area" in calculus? I'm doing homework dealing with definite integrals and some problems ask for area and some ask for net area.

Thank you for your help in advance. 😀

Here is a reasonable video of what you want.

https://www.youtube.com/watch?v=ghUVDEdtdpM

I don't like the way he abuses the equal sign and slaps his arithmetic all over the place
but the problems illustrates what happens when some of the area comes out as negative.

If that happens, just sketch the curve and break the integral into separate parts, subtracting the negative areas if necessary.

Thank you, I watched it and it helped me quite a bit. However, I'm now concerned that a problem like this will be on my final exam next week where I am not allowed to use a graphing calculator.

Back in the olden days, we did not have calculators of any kind.

As oobleck stated, it is a good idea to always make a sketch to have
a mental image of what the problem is.
Look if part of the graph is above and part is below the x-axis within your
integral boundaries. Often you will have to find the x-intercepts to see where
you must split your integrals. In the video I gave you, it was obvious that x = 1 was
the split.

You're welcome! I'd be happy to help explain the difference between "area" and "net area" in calculus.

In calculus, the term "area" typically refers to the concept of finding the absolute value of the area between a curve and the x-axis over a given interval. This is often represented using definite integrals. When finding the area between a curve and the x-axis, any portion of the graph that lies below the x-axis (negative values) is treated as positive for the purpose of calculating the area. Therefore, the calculated value gives the total area, regardless of the positive or negative regions.

On the other hand, "net area" considers the signed area between a curve and the x-axis. This means that portions of the graph that lie below the x-axis (negative values) will contribute to the calculated area as negative values. As a result, the net area can be negative, positive, or zero, depending on the regions above and below the x-axis.

To compute the area or net area using definite integrals, you need to set up an appropriate integration scheme. Here's a step-by-step guide to finding the area or net area:

1. Identify the interval over which you want to determine the area or net area.
2. Determine the function that represents the curve you are working with.
3. Sketch the graph of the function and determine the regions above and below the x-axis.
4. If you need to find the area, take the absolute value of the integrand when setting up the definite integral. The integrand will represent the function for the area.
5. If you need to find the net area, use the integrand as it is, without taking the absolute value.
6. Evaluate the definite integral using the appropriate integration bounds (start and end points of the interval).
7. The resulting value will give you the area or net area.

Remember, area considers all regions as positive, while net area considers regions above the x-axis as positive and regions below the x-axis as negative.