Integration with temperature and distance

[Hank.Shraeford]

Evening mates, I am in a bit of a quarrel here. Any help with this here question would be much appreciated. My Calculus class made me realize I am bloody terrible at math because I can't solve much of anything in this problem. Cheers!!

[attachment=0:1jqbcqe5]problem.jpg[/attachment:1jqbcqe5]

- To do A, I need to know T(x), and I am not sure how to find it. I wanted to try finding the slope, but I do not believe it is linear (if that matters)?
- Im still searching on the internet about how to do trapezoidal sums so I may be able to do this one once I read about it (my professor is horrid and has never taught us this)
- C should be rather simple once I know the function T(x) and find its derivative
- Once I know T '(x) I will just do the derivative once more and check the graph, so this part will be easy.

All in all it appears that I just need the most help with findind T(x) and knowing how to do trapezoidal sums. Should be able to take it from there.

 

[BigGlenntheHeavy]

$\displaystyle The \ Trapezoidal \ Rule:$

\(\displaystyle Let \ f \ be \ continuous \ on \ [a,b]. \ The \ Trapezoidal \ Rule \ for \ approximating \ \int_{a}^{b}f(x) \ dx \ is \\)

$\displaystyle given \ by$

$\displaystyle \int_{a}^{b}f(x) \ dx \ \dot= \ \frac{b-a}{2n}[f(x_0)+2f(x_1)+2f(x_2)+...+2f(x_{n-1})+f(x_n)].$

$\displaystyle Moreover, as \ n \ \implies \ \infty, \ the \ right-hand \ side \ approaches \ \int_{a}^{b}f(x) \ dx.$

 

[Hank.Shraeford]

what does n equal?

I think it would be [8/2n]*((100)+2(93)+2(70)+2(62)+(55)) correct? I just do not know what my n represents

 

[BigGlenntheHeavy]

$\displaystyle Hey \ mate, \ if \ you \ don't \ know \ what \ n \ equals, \ you're \ over \ your \ head.$

$\displaystyle By \ any \ chance, \ have \ you \ been \ supplied \ with \ a \ book \ on \ Calculus, \ if \ so \ look \ in \ the \ index$

$\displaystyle (I'm \ assuming \ it \ has \ one) \ and \ under \ T \ find \ what \ pages \ the \ trapezoidal \ rule \ is \ on.$

 

[Hank.Shraeford]

I would assume it to be the number of terms, but I didn't know if it was something different in the Trapezoid Rule.

So then I guess my n value is 4 since the question says using the four subintervals. So I would do (8/8)*[(93)+2(70)+2(62)+(55)] which = 412 ...No that cannot be right..


Okay so I have found it in my book, and it uses the example of y=sin(x) and how to do it, and it makes perfect sense. But I have to be doing something wrong with mine. I am going to go ahead and post this but I may edit it soon with a new answer.

 

[BigGlenntheHeavy]

$\displaystyle \int_{0}^{8}T(x) dx \ \dot= \ \frac{8-0}{(2)(4)}[T(0)+2T(1)+2T(5)+2T(6)+T(8)]$

$\displaystyle Now, \ try \ again.$

 

[Hank.Shraeford]

Oh man! Duh! Because from one value to another is the sub interval i.e. it takes 5 values to make 4 subintervals. Okay that makes sense now.

Now do I need to make a graph/plot the points to find T '(x) or how do I go about that?

 

[BigGlenntheHeavy]

$\displaystyle Hey, \ finish \ up \ with \ b \ first, \ then \ you \ can \ muck \ up \ the \ rest.$

$\displaystyle "To \ do \ A, \ I \ need \ to \ know \ T(x), \ and \ I \ am \ not \ sure \ how \ to \ find \ it."$

$\displaystyle Your \ quote, \ not \ mine.$