Double Integral over General Region : Hass Section 13.2 - Problem 5

In summary, the problem involves finding the outer and inner integrals of a function, and the function is 3y^3 * e^(xy). The solutions manual uses integration by parts to find the inner integral, resulting in [3y^2 * e^(xy)] from zero to y^2.
  • #1
atrain77abc
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Homework Statement



Outer Integral: From zero to one dy
Inner Integral: from zero to y^2 dx

Function is: 3y^3 * e^(xy)


Homework Equations


None


The Attempt at a Solution



Have tried numerous u substitutions on e^(xy), but taking me nowhere. I am clearly doing something wrong. Assuming 3y^3 is a constant and does not need to be integrated when integrated with respect to x.

Solutions manual shows result of inner integral being [3y^2 * e^(xy)] from zero to y^2 - which appears to me that a y in the original 3y^3 simply disappeared! No idea how they are getting from 3y^3 to 3y^2 as the result of the first integration!

I am quite sure this is an easy problem and I am simply overlooking a very simple step.
 
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  • #2
You need to use integration by parts.
 

Related to Double Integral over General Region : Hass Section 13.2 - Problem 5

What is a double integral over a general region?

A double integral over a general region is a mathematical operation that calculates the volume under a surface in three-dimensional space. It is represented by ∫∫f(x,y) dA, where f(x,y) is the function being integrated and dA represents the area element over the given region.

What is the difference between a general region and a rectangular region?

A general region is any two-dimensional shape, while a rectangular region is a specific type of general region with sides that are parallel to the x and y-axes. Double integrals over general regions require a different approach compared to those over rectangular regions.

How do you determine the limits of integration for a double integral over a general region?

To determine the limits of integration for a double integral over a general region, you need to analyze the given region and identify the boundaries of the shape. These boundaries will determine the limits for both the inner and outer integrals. The limits can also be expressed in terms of x or y, depending on which variable is being integrated first.

What is the purpose of using a double integral over a general region?

A double integral over a general region allows for the calculation of a volume in three-dimensional space. This is useful in many scientific and engineering applications, such as calculating the mass of an object with a non-uniform density or determining the center of mass of a complicated shape.

Are there any special cases or exceptions when working with double integrals over general regions?

Yes, there are special cases and exceptions when working with double integrals over general regions. Some common examples include regions with curved boundaries, regions with holes, and regions that cannot be easily divided into smaller shapes. In these cases, a different approach or additional techniques may be needed to solve the integral.

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