How do I integrate e^(y^3) in a definite integral without using erf(x)?

In summary, the individual is trying to integrate the function \int^3_0 \int^1_{\sqrt{\frac{x}{3}}} e^y^3,dydx and is having trouble with the inner integral \int^1_{\sqrt{\frac{x}{3}}} e^y^3 dy. They have looked for help on a forum and someone suggested using the error function (erf(x)), but they do not understand it. They are then advised to change the order of integration and given the integral \int_{y= 0}^1 \int_{x= 0}^{3y^2} e^{y^3} dx dy as a solution. This method works for them and
  • #1
exidez
44
0
The function i need to integrate is

[tex]\displaystyle\int^3_0 \int^1_{\sqrt{\frac{x}{3}}} e^y^3,dydx[/tex]

however my problem is integrating:
[tex]\int^1_{\sqrt{\frac{x}{3}}} e^y^3 dy[/tex]

I have looked on the forum and everyone is mentioning erf(x) which i don't understand yet. Do i need this for this definite integral? If so where can i learn about it and how to use it?
 
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  • #2
Try changing the order of integration.

For this integral, x ranges from 0 to 3 and, for each x, y from [itex]\sqrt{x/3}[/itex].

When x= 3, [itex]y= \sqrt{x/3}= 1[/itex] and when x= 0, [itex]y= \sqrt{0/3}= 0[/itex] so y ranges from 0 to 1. For each y, since [itex]y= \sqrt{x/3}[/itex] leads to [itex]y^2= x/3[/itex] or [itex]x= 3y^2[/itex], x ranges from 0 to 3y^2. Try
[tex]\int_{y= 0}^1 \int_{x= 0}^{3y^2} e^{y^3} dx dy[/tex]
 
  • #3
that actually worked out quite nicely. Thank you.
 

Related to How do I integrate e^(y^3) in a definite integral without using erf(x)?

1. What is the purpose of integrating e^(y^3)?

The purpose of integrating e^(y^3) is to find the area under the curve of the function e^(y^3). This can be useful in various fields of science, such as physics, chemistry, and biology, where the integration of a function can help in solving real-world problems.

2. What is the mathematical notation for integrating e^(y^3)?

The mathematical notation for integrating e^(y^3) is ∫e^(y^3)dy, where the integral symbol ∫ represents the process of integration and dy represents the variable of integration.

3. What are the methods for integrating e^(y^3)?

There are several methods for integrating e^(y^3), including substitution, integration by parts, and using special integration techniques such as trigonometric substitution or partial fractions. The appropriate method to use depends on the form of the function and the problem at hand.

4. Can e^(y^3) be integrated analytically?

No, e^(y^3) cannot be integrated analytically. This means that there is no closed-form solution for the integration of this function. However, it can still be integrated numerically using various numerical methods, such as Simpson's rule or the trapezoidal rule.

5. What are some applications of integrating e^(y^3)?

The integration of e^(y^3) has various applications in science, such as in calculating the work done by a force in physics, determining the rate of a chemical reaction in chemistry, and finding the growth rate of a population in biology. It can also be used in probability and statistics to find the area under a probability distribution function.

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