jonroberts74 said:Homework Statement
Set up the double integral over the region ##y=x+3; y=x^2+1##
Homework Equations
The Attempt at a Solution
finding the intersections you get the double integral
##\int_{1}^{5}\int_{-1}^{2}dxdy =12 ##
but why is that not the same as
##\int_{1}^{5}\int_{-1}^{\sqrt{y-1}} dxdy =\frac{28}{3}##
jonroberts74 said:yeah I drew it, its the line y=x+3 slicing through the parabola x^2 moved up one unit on the y-axis
##\int_{1}^{5}\int_{-\sqrt{y-1}}^{\sqrt{y-1}} dxdy ## is the other one I had, but no sure if that is correct either
LCKurtz said:It isn't. If you do x first it must go from ##x_{left}## to ##x_{right}##. x on the left is a two piece formula.
Is there some reason you don't do this problem the more natural and easier dydx order?
jonroberts74 said:probably out of habit of typically slicing with a y value then seeing how x changes
##\displaystyle\int_{-1}^{2}\int_{x^2+1}^{x+3} dydx##
okay I see it now, if I do x first there's an issue where the left intersection point occurs, below that point the parabola is the left and right x and above that the left is the line and the right is the parabola.
thanks!
##\displaystyle \int_{1}^{2} \int_{-\sqrt{y-1}}^{\sqrt{y-1}}dxdy + \int_{2}^{5} \int_{y-3}^{\sqrt{y-1}}dxdy = \frac{4}{3} + \frac{19}{6} = \frac{9}{2}##LCKurtz said:Yes! That's why you need two integrals to do it it dxdy order. That's the sort of thing you should always look at when deciding how to set up a double integral. Now can you get the same answer both ways?
A double integral is a mathematical concept that involves calculating the area under a two-dimensional surface. It is useful because it allows us to find the volume or area of irregular shapes that cannot be easily calculated using traditional methods.
To set up a double integral, you need to define the boundaries of the region you want to find the area or volume of, and then choose the appropriate function to integrate over that region. To solve the double integral, you can use various techniques such as iterated integrals, change of variables, or polar coordinates.
Double integrals are used in many fields such as physics, engineering, economics, and computer graphics. Some examples include calculating the center of mass of an object, finding the work done by a force, determining the area of a region on a map, and creating 3D models of objects.
A single integral involves calculating the area under a one-dimensional curve, while a double integral involves calculating the area under a two-dimensional surface. This means that a single integral has one variable, while a double integral has two variables.
One common mistake is forgetting to properly define the boundaries of the region to be integrated. Another is not taking into account the orientation of the region, which can lead to incorrect results. It is also important to choose the correct order of integration, as switching the order can change the final result.