# Definite Integrals

1. Apr 27, 2007

### Ry122

1.5 to -1.5
6e^(3x)
end up with 2e^4.5 - 2e^-4.5 (Found indefinite integral and sub'd 1.5 with x etc)
The answer according to the text book is
180.0342 - .0222 Im lost from this part. I dont know how 2e^4.5=180.0342
=180.01

2. Apr 27, 2007

### Gib Z

u=3x
du=3 dx
$2\int e^u du = 2(e^{3x}) + C.$ Now sub in the bounds.

3. Apr 27, 2007

### Ry122

What are the bounds?
Does something have to substituted for e in 2e^4.5 to get 180.0342?

4. Apr 27, 2007

### Gib Z

Once we have the anti derivative, F(x) of a function f(x), to find the area we subtract F(b) - F(a) where b is the top of the integration and b the bottom. So this time b=1.5, a = -1.5

5. Apr 27, 2007

### D H

Staff Emeritus
Because that is what it evaluates to. Do you know what $e^a$ means?

From this post, it appears you have a basic misunderstanding of the exponential function. The wikipedia article, http://en.wikipedia.org/wiki/Exponential_function, is fairly well written.

If you have a scientific calculator, you can use it to calculate $2e^{4.5}$. Almost every scientific calculator has an exponential function button. It is labeled either $e^x$ or $\exp$. Simply enter 4.5, press the exponential function button, and multiply by 2.

Last edited: Apr 27, 2007
6. Apr 27, 2007

### HallsofIvy

Staff Emeritus
If you are able to integrate ex then surely you understand that "e" is just the base of natural logarithms, approximately 2.71i. To find e4.5, use a calculator for goodness sake!

7. Apr 27, 2007

### Ry122

After reading that last post I only just found out that e is a constant. What is the reason for having a constant in calculating logs?

8. Apr 27, 2007

### D H

Staff Emeritus
It really does appear you have a fundamental misunderstanding of the concept of the exponential function and its inverse, the natural logarithm.

You know what 10*10, 10*10*10, 10*10*10*10, etc., are. Another way to write those numbers is to use the exponentiation operator: 10*10=102, 10*10*10=103, and so on. Just like multiplication by an integer is nothing more than repeated addition, raising a number to an integral power is nothing more than repeated addition. And just as multiplication can be extended to the reals, so can exponentiation. Extending exponentiation to the reals leads to functions like f(x)=10x.

There is nothing special about 10. One can similarly calculate things like 2/32, 2/33, and more generally, 2/3x. e is just another number in this regard. ex is just another exponential function like 10x. The function f(x)=ex has such incredible properties that it is called the exponential function.

Functions can have inverses. The inverse of a function is not f-1(x)=1/f(x). Instead, it is the function that "undoes" what the function did: f-1(f(x))=x. For example, the inverse of f(x)=x2 is the square root function. The exponential function has an inverse as well, the natural logarithm function.

9. Apr 28, 2007

### HallsofIvy

Staff Emeritus
I find it difficult to believe that you are integrating functions but have never seen a definition of ex or (another name for the same function) exp(x). Surely, you have already learned the derivative of ex?