Calculating the Area Between Two Curves | A Step-by-Step Guide

area between 2 curves (for functions of y)

The area between two curves refers to the region enclosed by two curves on a coordinate plane

The area between two curves refers to the region enclosed by two curves on a coordinate plane. When dealing with functions of y, we are interested in finding the area between two curves in terms of y-values. To calculate this area, we follow these steps:

1. Identify the two curves: Let’s assume we have two functions, f(y) and g(y), which represent the upper and lower curves respectively. These functions could be given explicitly, such as f(y) = 4y and g(y) = y^2, or implicitly as equations involving y.

2. Determine the y-values for the region of interest: Find the values of y where the upper and lower curves intersect. This can be achieved by setting the two functions equal to each other and solving for y. Let’s denote the y-values of intersection as y1 and y2, where y1 < y2. 3. Determine the limits of integration: Decide on the interval over which we want to calculate the area. This interval is determined by the lower and upper limits of y and is usually from y1 to y2. 4. Set up the integral: The area between two curves can be found by integrating the difference between the upper curve (f(y)) and lower curve (g(y)) with respect to y over the given interval. The integral can be expressed as: A = ∫[g(y), f(y)] (f(y) - g(y)) dy Note: Make sure to choose the correct order when subtracting the functions, as it should be upper curve minus lower curve. 5. Evaluate the integral: Use appropriate integration techniques to evaluate the integral. This could involve using basic rules of integration, substitution, or partial fractions, depending on the complexity of the functions. 6. Calculate the area: Once the integral is evaluated, the result will give the area between the two curves. Make sure to consider the positive value of the integral as the area cannot be negative. It's important to note that the area between two curves can also be computed in terms of x-values if the functions are given in terms of x. In that case, we would follow a similar process, but with the variables and limits adjusted accordingly.

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