Double Integral over B: Evaluate x^2y^3

• cse63146
In summary, the conversation is about evaluating a double integral over a closed region, where the region is bounded by two equations. There is some confusion about setting up the parameters correctly, but it is determined that the answer is 1/77. There is also discussion about taking the absolute value and determining which variable is larger in different scenarios.
cse63146

Homework Statement

Evaluate the following double integral over the closed region B:

$$\int\int_B x^2 y^3 dx dy$$ where B is the region bounded by y = x^2 and y = x

The Attempt at a Solution

I think I set up the paramaters wrong:

$$\int^{1}_{0}\int^{y}_{\sqrt{y}} x^2 y^3 dx dy$$

Last edited:
If y is in [0,1], as it looks like you've figured out for your integral, which is larger y or sqrt(y)? Is that what's confusing you?

y>sqrt(y)

The answer is 1/77, but I get -1/77, but taking it's absolute value should take care of that.

cse63146 said:
y>sqrt(y)
What's sqrt(1/4)?

cse63146 said:
y>sqrt(y)

The answer is 1/77, but I get -1/77, but taking it's absolute value should take care of that.
Can you explain why "taking its absolute value" is justified?

Once again, which is larger, y or $\sqrt{y}$ for y between 0 and 1?

Which is larger 1/4 or $\sqrt{1/4}$?

I get it. Because y goes from 0 - 1, sqrt(y)>y in this case.

but what if y went from 0 - 2, then would it be this:

$$\int^{1}_{0}\int^{\sqrt{y}} }_{y} x^2 y^3 dx dy + \int^{2}_{1}\int^{y}_{\sqrt{y}} x^2 y^3 dx dy$$

Yes.

Thank you all for your help.

1. What is a double integral?

A double integral is a type of mathematical calculation used in multivariable calculus to determine the volume under a surface in a three-dimensional coordinate system. It involves solving two integrals, one after the other, to find the total volume.

2. How do you evaluate a double integral?

To evaluate a double integral, you must first set up the integral using the given function and the limits of integration. Then, you can either use the integration rules to solve the integral or use a graphing calculator or computer program to find the numerical answer. In this case, we will use the integration rules to solve the double integral of x^2y^3.

3. What does the notation "x^2y^3" mean?

The notation "x^2y^3" means that the given function is a monomial, which is a single term algebraic expression. The x^2 and y^3 represent the variables and their respective exponents. In this problem, we are evaluating the double integral of this monomial over a specific region B.

4. What is the purpose of evaluating a double integral?

The purpose of evaluating a double integral is to find the volume under the surface of a three-dimensional region. This is useful in many scientific and mathematical applications, such as calculating the volume of a solid object or determining the mass of an irregularly shaped object.

5. Can a double integral be evaluated over any region?

No, a double integral can only be evaluated over regions that are bounded by a continuous curve or surface. In this problem, we are evaluating the double integral over the region B, which is a specific region defined by its limits of integration. It is important to carefully determine the appropriate region for a double integral before attempting to solve it.

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