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Applications of Area under a Curve

at November 5th 2023, 7:25:26 pm.

Post 5: Applications of Area under a Curve

The concept of finding the area under a curve has numerous applications in various fields. It allows us to measure quantities, analyze trends, solve real-world problems, and make predictions. In this post, we will explore a few applications to better understand the significance and practical uses of calculating the area under a curve.

1. Physics

In physics, the area under a curve represents the displacement or distance traveled by an object with respect to time. If the curve represents the velocity of an object over a certain time interval, the area under the curve will give us the total displacement of the object during that interval.

For example, let's consider a car's velocity over a 10-second interval given by the function:

v(t) = 2t

To find the total displacement of the car during this time, we need to calculate the area under the curve of $v(t)$ from $t = 0$ to $t = 10$ :

\text{Displacement} = \int_{0}^{10} 2t \,dt

Evaluating the definite integral, we find:

\text{Displacement} = [\frac{t^2}{2}]_{0}^{10} = \frac{10^2}{2} - \frac{0^2}{2} = 50 - 0 = 50

Hence, the car traveled a total distance of 50 units during the 10-second interval.

2. Economics

In economics, the area under a curve can represent economic measures such as consumer surplus, producer surplus, and market equilibrium. These measures help economists analyze supply and demand, market efficiency, and the allocation of resources.

For instance, consider the supply and demand curves for a certain product:

Supply and Demand Curves

The area above the market price and below the demand curve represents consumer surplus, which represents the extra benefit consumers receive by purchasing the product at a price lower than what they are willing to pay.

Similarly, the area below the market price and above the supply curve represents producer surplus, indicating the extra benefit producers receive by selling the product at a price higher than their production cost.

3. Biology

In biology, the area under a curve can depict the growth or decay of populations, the concentration of a drug in the body over time, or the accumulation of a certain substance.

For example, let's consider the growth of a bacterial population given by the function:

P(t) = 100e^{0.1t}

The area under the curve of $P(t)$ from $t = 0$ to $t = 10$ can represent the total number of bacteria at a given time interval.

By evaluating the definite integral, we find:

\text{Total bacteria} = \int_{0}^{10} 100e^{0.1t} \,dt

Solving this integral, we obtain:

\text{Total bacteria} = \left[10e^{0.1t}\right]_{0}^{10} = 10e^{0.1 \cdot 10} - 10e^{0.1 \cdot 0} \approx 271.83

Hence, there would be approximately 271.83 bacteria after 10 units of time.

These are just a few examples showcasing the diverse applications of finding the area under a curve. It's worth noting that this concept is not limited to these fields but can be applied in countless other areas such as engineering, medicine, and environmental sciences, to name a few.