My book says to do the following problem via computer and via hand:
Calculate the definite Integral of
e^-x * cos x dx
over (0, +infinity)
My TI-89 calculator gets 1 (it gets the same thing when I replace infinity with 999).
when I do this by hand, I get: 1/2.
The formula e^-x * cos x dx integrates to:
1/2 * e^-x * (sin x - cos x)
calculating:
limit b->infinity: 1/2 * e^-b * (sin b - cos b) - 1/2 * e^0 * (sin 0 - cos 0)
= 0 - 1 * (0 - 1/2) = 1/2
1/2 matches the book's answer. My question is why does my TI-89 calculator return the wrong answer of 1? I assume the text book devised this problem to exploit some computer weakness?
Perhaps the calculator is doing a numerical integration and the components are going +-+-+-+- with the phase of the cosine and it is just asking too much of the poor little calculator :)
In other words, I have no idea.
That's probably close enough Damon. thanks!
It seems like there might be a misunderstanding in your approach to using the calculator to evaluate the definite integral. Let's go over the correct process step by step.
To evaluate the definite integral using a calculator, you need to find a numerical approximation of the integral over a finite interval. In this case, the bounds of integration are from 0 to infinity. Since infinity is not a number that can be input into the calculator, it is necessary to choose a sufficiently large upper limit and then examine the results as the limit tends towards infinity.
Using the TI-89 calculator, you can evaluate the integral as follows:
1. Press the "Apps" button and select the "Calculus Made Easy" application.
2. Choose "Definite Integral" from the menu.
3. Enter the function as "e^(-x)*cos(x)".
4. Specify the variable as "x".
5. Set the lower limit as "0".
6. Choose a sufficiently large upper limit, such as "999".
7. Press "Enter" to calculate the approximation.
Now, when you evaluate this integral with the above steps, the calculator should return an approximation. However, it's important to note that since the upper limit is not truly infinity, the result will only be an approximation. Your calculator might display 1, but it's not necessarily the exact value of the integral over the infinite interval.
On the other hand, your manual calculation using the limit points out the correct approach. By taking the limit as the upper bound tends towards infinity, you obtained the correct result of 1/2. This manual calculation accounts for the actual behavior of the function as x approaches infinity and provides a more accurate result.
In conclusion, the discrepancy between the calculator's approximation and the manual calculation arises from the limitations of numerical approximations and the chosen upper limit. Always keep in mind that numerical methods might have limitations, and manual calculations can often provide more precise results, especially in cases like evaluating integrals over an infinite interval.