flyers said:Homework Statement
Find the area bounded by the curves given by the graphs x = 4 - y^2 and x = y^2 - 2y
The Attempt at a Solution
I don't know how I can integrate these curves as they are not functions. Can someone tell me to get started on this problem?
Thank you
They aren't functions, but they are curves, and they define a bounded region. Start by graphing them and finding where the two curves intersect, and then figure out what your typical area element looks like - i.e., horizontal strip of vertical strip- and its dimensions.flyers said:Homework Statement
Find the area bounded by the curves given by the graphs x = 4 - y^2 and x = y^2 - 2y
The Attempt at a Solution
I don't know how I can integrate these curves as they are not functions. Can someone tell me to get started on this problem?
Thank you
Mark44 said:They aren't functions, but they are curves, and they define a bounded region. Start by graphing them and finding where the two curves intersect, and then figure out what your typical area element looks like - i.e., horizontal strip of vertical strip- and its dimensions.
True, but people generally think of functions where y is the dependent variable and x is the independent variable, out of force of habit. In that sense, the equations don't represent functions.Char. Limit said:Actually, they are functions. They're just functions of y, not the functions of x most people are used to.
The area between curves refers to the region bounded by two or more curves on a graph. This area can be found by calculating the definite integral of the difference between the two curves over a given interval.
To find the bounded region between two curves, you will need to first graph the two curves and determine the points of intersection. Then, set up the integral by subtracting the lower curve from the upper curve and integrating over the interval of the points of intersection.
The formula for finding the area between curves is given by the definite integral of the difference between the two curves over a given interval. This can be written as ∫(f(x) - g(x)) dx, where f(x) and g(x) are the two curves and dx represents the integration with respect to x.
Yes, the area between curves can be negative if the lower curve is above the upper curve in certain parts of the interval. This means that the definite integral will result in a negative value, indicating that the bounded region has a negative area.
Finding the area between curves has many real-life applications, such as calculating the total profit or loss in a business, determining the amount of material needed for a construction project, and predicting the growth or decline of a population. It can also be used in physics to calculate the work done by a varying force or to find the area under a velocity-time graph.