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Area Under a Curve
Calculating the area under a curve.
Definite Integrals
So far when integrating, there has always been a constant term left. For this reason, such integrals are known as indefinite integrals. With definite integrals, we integrate a function between 2 points, and so we can find the precise value of the integral and there is no need for any unknown constant terms [the constant cancels out].
The Area Under a Curve
The area under a curve between two points can be found by doing a definite integral between the two points.
To find the area under the curve y = f(x) between x = a and x = b, integrate y = f(x) between the limits of a and b.
Areas under the x-axis will come out negative and areas above the x-axis will be positive. This means that you have to be careful when finding an area which is partly above and partly below the x-axis.
You may also be asked to find the area between the curve and the y-axis. To do this, integrate with respect to y.
Example
Find the area bounded by the lines y = 0, y = 1 and y = x2.
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FAQs on Finding areas using integration, Business Mathematics & Statistics - Business Mathematics and Statistics - B Com
| 1. What is integration and how is it used to find areas? |  |
Ans. Integration is a mathematical technique that allows us to find the area under a curve or between two curves. By integrating a function, we can calculate the total area enclosed by the curve(s) and the x-axis. This is achieved by breaking the region into infinitesimally small rectangles and summing up their areas using the definite integral.
| 2. Can integration be used to find areas in business and economics? | |
Ans. Yes, integration is extensively used in business and economics to calculate areas and solve various problems. For example, it can be used to determine the area under a demand curve to find the total revenue or to calculate the area between a supply curve and the equilibrium price to find the producer surplus.
| 3. Are there any limitations to using integration for finding areas in real-world scenarios? | |
Ans. While integration is a powerful tool for finding areas, there are some limitations in real-world scenarios. One limitation is that integration assumes a continuous and well-behaved function, which may not always accurately represent real-world data. Additionally, practical constraints such as limited data points or non-linear relationships can make integration challenging to apply directly.
| 4. Can integration be used to find areas of irregular shapes or regions? | |
Ans. Yes, integration can be used to find areas of irregular shapes or regions. By breaking down the irregular shape into smaller, manageable pieces, we can integrate the functions that describe each piece to find their individual areas. Summing up these areas using the definite integral allows us to calculate the total area of the irregular shape.
| 5. How does integration relate to business decision-making and optimization? | |
Ans. Integration plays a crucial role in business decision-making and optimization. By using integration techniques, businesses can analyze and optimize various aspects such as production costs, profit margins, and resource allocation. Integration helps in determining optimal production levels, pricing strategies, and maximizing profits by finding areas that represent relevant economic measures in these decision-making processes.
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