Definite Integrals of 6e^(3x) from 1.5 to -1.5: Solution and Explanation

[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 textbook is
180.0342 - .0222 I am lost from this part. I don't know how 2e^4.5=180.0342
=180.01

 

[Gib Z]

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

 

[Ry122]

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

 

[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

 

[D H]

I don't know how 2e^4.5=180.0342

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

Does something have to substituted for e in 2e^4.5 to get 180.0342?

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.

Edited to add

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.

 

[HallsofIvy]

If you are able to integrate e^x then surely you understand that "e" is just the base of natural logarithms, approximately 2.71i. To find e^4.5, use a calculator for goodness sake!

 

[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?

 

[D H]

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)=10^x.

There is nothing special about 10. One can similarly calculate things like 2/3², 2/3³, and more generally, 2/3^x. e is just another number in this regard. e^x is just another exponential function like 10^x. The function f(x)=e^x 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.

 

[HallsofIvy]

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?

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