# Help for Tom's Daughter with Integration of 0 to Infinity

• Tompman
In summary, the conversation discusses a mathematical problem involving integration from 0 to infinity. The problem is related to reserves of oil in the world and initially appears to be unsolvable analytically, but can be approximated using certain techniques. The final answer, according to Maple, is approximately 28.48936147.
Tompman
Hi,

Can anyone please help me re integration from 0 to infinity of the following;

(e^0.03t)((50e^0.07t-10)/k)^-0.5 dt

Yours sincerely,

Tom.

Doing that problem analytically is probably not possible, but if we very vaguely approximate the integrand, we can see what it is for most practical purposes.

Your integral is simplifies to $$\sqrt{\frac{k}{50}} \int^{\infty}_0 \frac{e^{0.03t}}{e^{0.07t} - 0.2} dt$$.

The next step takes away a lot of accuracy, but its the best i can really think of that doesn't take forever to do. We pretend we don't see the -0.2 in the denominator lol.

If we do that and evaluate the integral, we should get $$5\sqrt{ \frac{k}{2} }$$.

What class is your daughter in that she would have a problem like that?

Gib Z,
Thank you. I have sent this to my daughter.

HallsofIvy,
My daughter is doing economics in university and this has something to do with the reserves of oil in the world.

Using maple the integral

$$\int^{\infty}_0 \frac{exp(0.03t)}{exp(0.07t) - 0.2}dt$$

is approximately 27.07839303

nicksauce said:
Using maple the integral

$$\int^{\infty}_0 \frac{exp(0.03t)}{exp(0.07t) - 0.2}dt$$

is approximately 27.07839303

O that's pretty good then =] When we ignored that -0.2, it was 25.

Am I missing something, or shoudn't the integral portion be

$$\int^{\infty}_0 \frac{e^{0.03t}}{\sqrt{e^{0.07t} - 10}}dt$$

Yes Theo, having spoken to my daughter I think you are correct. What would the answer be then? All replies are very welcome.

According to maple, the integral

$$\int_0^{\infty}\frac{exp(0.03t)}{\sqrt{50exp(0.07t)-10}}$$

is approximately 28.48936147

TheoMcCloskey said:
Am I missing something, or shoudn't the integral portion be...

Sorry my bad >.<" Maybe that's why I was so surprised that my 'approximation' was quite close, because I only ignored 0.2 instead of a 10. If we try to ignoring thing again I'm sure it would be quite a bit off.

According to maple, the integral ... is approximately 28.48936147

I'm still looking at this, but I'm not done. However, I don't think the value is that high. Initially, I'm getting something roughly half this value.

more to come.

After further review, I indeed get results that agree with nicksauce, ie, The integral is approx equal to 28.489361...

## 1. What is integration of 0 to infinity?

Integration of 0 to infinity, also known as improper integration, is a mathematical process that involves finding the area under a curve from 0 to infinity. It is a type of indefinite integration that is used to solve problems involving infinite limits.

## 2. Why is integration of 0 to infinity important?

Integration of 0 to infinity is important because it allows us to solve problems that involve infinite limits and can be used to find the total area under a curve. It is also used in various fields of science and engineering, such as physics, economics, and statistics.

## 3. How is integration of 0 to infinity performed?

Integration of 0 to infinity is performed using the same techniques as regular integration, such as u-substitution, integration by parts, and trigonometric substitution. However, the limits of integration are taken to infinity, which requires some additional steps to ensure the convergence of the integral.

## 4. What are the challenges of integration of 0 to infinity?

One of the main challenges of integration of 0 to infinity is ensuring the convergence of the integral. This requires careful selection of the method of integration and the use of techniques such as limit comparison and comparison test to determine the convergence of the integral.

## 5. What are some real-life applications of integration of 0 to infinity?

Integration of 0 to infinity has many real-life applications, such as calculating the total revenue or cost in economics, finding the total energy or power in physics, and determining the probability of an event in statistics. It is also used in various engineering fields to solve problems involving infinite limits.

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