Discussion: Net area vs Total area What is the difference between net area and total area from the curve to the x-axis? What is the difference in using definite integrals to calculate the total area from the curve to the x-axis from x = a to x = b rather than the net area? What would the solutions to integral(s) calculating the net area as well as total area of a velocity function tell you about the situation? Include examples in your detailed explanations. You cannot use an example that has been posted by someone else. You need to post something original. You are also required to critique or question another student’s post. I will withhold my comments for a period of time allowing you ample time to do so.
The Correct Answer and Explanation is:
Net Area vs. Total Area
Net area refers to the area under a curve, considering both the positive and negative portions. In this context, the area above the x-axis is positive, while the area below the x-axis is negative. The total net area is found by evaluating the definite integral of a function over a given interval, and this integral includes contributions from both positive and negative values of the function.
Total area, on the other hand, refers to the total space between the curve and the x-axis, without considering whether the curve is above or below the axis. This area is always non-negative, and we find it by taking the absolute value of the function where it is below the x-axis before integrating. Essentially, this method gives us the total positive area, regardless of the function’s direction.
Difference in Using Definite Integrals
To calculate the net area from a curve between x = a and x = b, we simply evaluate the definite integral of the function f(x) from a to b:Net Area=∫abf(x) dx\text{Net Area} = \int_a^b f(x) \, dxNet Area=∫abf(x)dx
This integral considers both positive and negative areas. If the function has negative values over part of the interval, it subtracts that area, potentially yielding a negative result.
However, to calculate the total area, we use the absolute value of the function:Total Area=∫ab∣f(x)∣ dx\text{Total Area} = \int_a^b |f(x)| \, dxTotal Area=∫ab∣f(x)∣dx
This ensures that the areas under the curve, regardless of direction, are counted as positive.
Application to Velocity Functions
When dealing with a velocity function, net area gives us the displacement of an object over a time period. If the velocity is negative over a portion of the interval, the displacement is negative, indicating movement in the opposite direction. The total area under the velocity curve, however, would give us the total distance traveled, treating all motion as positive regardless of direction.
For example, consider the velocity function of an object:v(t)=10−tfor0≤t≤10v(t) = 10 – t \quad \text{for} \quad 0 \leq t \leq 10v(t)=10−tfor0≤t≤10
To calculate the net area (displacement), we integrate directly:∫010(10−t) dt=[10t−t22]010=100−50=50\int_0^{10} (10 – t) \, dt = [10t – \frac{t^2}{2}]_0^{10} = 100 – 50 = 50∫010(10−t)dt=[10t−2t2]010=100−50=50
This means the object has displaced 50 units in the positive direction.
To calculate the total area (total distance), we must account for any negative velocity:∫010∣10−t∣ dt\int_0^{10} |10 – t| \, dt∫010∣10−t∣dt
For this function, 10 - t is positive for t < 10 and negative for t > 10, so we would adjust the integration accordingly. The total area would be the sum of the positive areas, regardless of the sign of the velocity.
Critique or Question:
I see you’ve used an example involving acceleration and distance. While it’s a good example, I wonder if there’s a case where the velocity function changes direction multiple times during the interval. How would this affect the net vs. total area, and how would you calculate the total distance if the direction of motion keeps changing?
