The double integral is a fundamental concept in calculus. It refers to the idea of calculating the area under a curve and above a boundary or displacement.
Unlike single integrals, which only require the ability to calculate an area, double integrals require the ability to calculate two areas!
Calculating a double integral requires being able to: divide areas into thin strips, add those strips together, and re-arrange equations. Simple, right?
Just like with single integrals, there are many situations where you will not need to calculate the full double integral. For example, if you are given the area under a curve and asked to find the volume directly above that area, you will not need to calculate a double integral. You will only need to calculate a single integral in that case.
Understanding the notation
The double integral requires more complex notation than the single integral. The addition of the D indicates that this is a double integral, and the letters at the end stand for what is being integrated over.
The letter Da indicates which variable is being divided by, which in this case is y. The ∫ symbol remains the same, and just adds a 2 to indicate that there are two variables being integrated over instead of one.
The triple integral requires three variables to be integrated over, one of which must be axes coordinates. For example, if the variable being integrated over was water depth, then X would have to be length and Y would have to be height in order to integrate properly.
There are also situations where there is no variable to divide by, but there are additional dimensions that need to be integrated over.
Understanding the limits
The limits of the double integral depend on what the variable changes are in the variable ranges. The most common variables that change are x and y, which represent the horizontal and vertical dimensions, respectively.
When y is constant, or Y = X where X is the maximum value of y, then the integral is simply equal to an area. This is because you know the height of the area, which is Y = X, so you know how wide or long it is.
When x is constant, or X ≥ 0 then the integral is simply equal to a volume. This is because you know the width of the volume, so you know how deep it is.
These two situations can be combined to find a more general solution. If both x and y are constants then there are no changes in either dimension, so the integral is just an area.
Calculating the area of a triangle
A special case of the double integral is when the interval of integration is a triangle. In this case, the integral is calculated by finding the average area of the triangle sides, and then adding that to the area of the bottom side.
The average side length is found by dividing the length of one side by two. The length of one side is simply the width or length of the interval being integrated.
For example, if we wanted to find the area of a square with a width of two units, we would have to integrate from zero to two units. The average side length would be one unit, which would be added to one unit for the bottom side, making an area of two units.
This proves that the general formula for integrating an n-sided figure with an integration interval that is twice as long as one side is equal to 2n−2.
Calculating the area of a square
A simple way to calculate the area of a square is by using the length of one side multiplied by the length of the other side.
A × B = Area of Square where A is one side and B is the length of the other side
In mathematics, there are many ways to calculate an area. One such way is by using a double integral.
A double integral calculates the area under a curve and above a plane that surrounds the curve. The process of calculating a double integral requires several steps and general guidelines on how to do so.
There are three cases for determining whether or not an integral can be converted to a double integral, one of which being when the region bounded by the function y = x is bounded above by y = X.
Evaluating double integrals
Once you have determined that a region is bounded by y = x, y = x3, x ≥ 0 and A(x,y)dx + B(x,y)dy ≤ C, you can evaluate the double integral.
First, rewrite the integral as:
Then change the order of integration:
Now break down the integration into two separate integrations: one for A(x,y)dx and one for B(x,y)dy. Using the fundamental theorem of calculus (that integration and differentiation are inverse operations), you can then integrate both sets.
The first set is integrated by taking the derivative of both sides with respect to x and then integrating this value with A(x,y)dx. The same procedure is done for B(x,y)dy.
Understanding symmetry with respect to axes
Symmetry with respect to axes is an important concept in geometry. You can apply this concept to the integral domain as well.
The double integral domain consists of all values of (x,y) such that (x,y) ∈ R2 × R. The boundaries of the double integral domain are x = 0 and y = 0.
Symmetry with respect to the x-axis means that if you take any point (x,y) in the plane and switch the x coordinate with 0, then the point will still be in the domain. The same goes for symmetry with respect to the y-axis.
The double integral domain is not bounded by z=0 because if you took any point in the plane and set z=0, then the point would not be in the domain.
Double integrals and volumes
A double integral is used to calculate the volume of a solid shape. The volume of a shape is how many units of measure it takes to fill that shape.
For example, if you poured enough water to fill a bowl, then the volume of the bowl is how many gallons, quarts, or liters it would take to fill it.
How do you evaluate a double integral? You first have to break it up into two separate integrals. The first integral is the integration of the outer region, and the second integral is the integration of the inner region. Then you evaluate each integral separately using established methods.
The hardest part about evaluating double integrals is determining what order to do the integrals in. There are rules for determining this, but they are not universal for all cases.
Converting from areas to volumes and vice versa
Once you are comfortable with the basic concepts of the definite integral, you can try your hand at more advanced uses. One of these is converting an area to a volume and vice versa.
If you have an area, as in the case of a rectangle, you can divide it into slices of equal thickness, and then sum up the volumes of all of those slices. This is how you would get a volume!
If you have a volume, then you can divide it into layers of equal height, and then sum up the areas of those layers. This is how you would get an area!
Both cases use the same concept: breaking down the shape into smaller pieces that have known values. By counting and calculating the total value of those pieces, you can find the total value of your shape.
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