Double Integral in Rectangular Coordinates

In summary, a double integral in rectangular coordinates is a mathematical tool used to calculate the volume between a two-dimensional region and the x-y plane. It involves integrating a function over a two-dimensional region by breaking it down into infinitely small rectangles and summing their areas. To calculate a double integral, one must first define the limits of integration and then integrate the function multiplied by an infinitesimal area element dA with respect to both x and y. This tool has many applications in physics, engineering, and other sciences, such as calculating areas, volumes, and centers of mass. However, it is limited to finding volumes of regions that can be represented by rectangles and cannot be used for irregular shapes or regions with curved boundaries. A double integral in rectangular
  • #1
memish
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Homework Statement



mcqm2r.jpg


Homework Equations



n/a

The Attempt at a Solution


I set up the intgral at
integral from 0 to 5 of integral from 0 to 5y of 8e^(y^2)dxdy

I solved it as an iterated integral so I solved the first part, then ended up with integral from 0 to 5 of 40ye^(y^2)

Am I going about it right? Isn't there no antideriative for e^(y^2)? I don't know where to go from here in solving this

Thanks everyone.
 
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  • #2
Yes, there's no elementary antiderivative for e^(y^2). There is one for y*e^(y^2). Use the substitution u=y^2.
 

What is a double integral in rectangular coordinates?

A double integral in rectangular coordinates is a mathematical tool used to calculate the volume between a two-dimensional region and the x-y plane. It involves integrating a function over a two-dimensional region by breaking it down into infinitely small rectangles and summing their areas.

How is a double integral in rectangular coordinates calculated?

A double integral in rectangular coordinates is calculated by first defining the limits of integration, which determine the bounds of the region being integrated. Then, the function being integrated is multiplied by an infinitesimal area element dA, and the resulting expression is integrated with respect to both x and y to find the total volume.

What is the significance of a double integral in rectangular coordinates?

A double integral in rectangular coordinates has many applications in physics, engineering, and other sciences. It can be used to calculate the area under a curve, the volume of a solid, the center of mass of an object, and many other quantities that involve two-dimensional regions.

What are the limitations of a double integral in rectangular coordinates?

A double integral in rectangular coordinates is limited to finding the volume of regions that can be represented by rectangles. It cannot be used to calculate volumes of irregular shapes or regions with curved boundaries.

How is a double integral in rectangular coordinates related to a single integral?

A double integral in rectangular coordinates is essentially the combination of two single integrals, one with respect to x and one with respect to y. The order in which these integrals are evaluated can affect the final result, but the overall process is similar to that of a single integral.

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