## ET365 one year ago Integral from 0,4 (5-2x)dx, guys whats the simplest way to work this??

1. Vocaloid

well, do you know how to calculate the antiderivative (integral) of 5 - 2x?

2. ET365

That would be x^2+c correct?

3. SolomonZelman

No

4. ET365

or rather 5-x^2|0,2?

5. SolomonZelman

I will give you an example that is similar to this: $$\large\color{black}{ \displaystyle \int_{ 2 }^{8}\left(7-10x\right){~}dx }$$ *[1]* An integral of a constant *b* (with respect to x) is equal to *bx*. (For any non-zero number *b*.) *[2]* An integral of any $$x^n$$ (with respect to x) is $$x^{n+1}/(n+1)$$, and integral of b•$$x^n$$ (when you have a coefficient b in front) is $${\rm b}x^{n+1}/(n+1)$$ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ So you first integrate each term: 1) The integral of 7 is 7x (based on *[1]* ). 2) The integral of -10x (which is same as -10x$$^1$$, is $${\rm -10}x^{1+1}/(1+1)=-10x^2/2=-5x^2$$ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ So you get: $$\large\color{black}{ \displaystyle \int_{ 2 }^{8}\left(7-10x\right){~}dx =7x-5x^2 }$$ and then evaluate that from x=2 to x=8 (those are the limits of integration) In other words: $$\large\color{black}{ \displaystyle \int_{ 2 }^{8}\left(7-10x\right){~}dx =\left[7(8)-5(8)^2\right] -\left[7(2)-5(2)^2\right]=-258 }$$

6. SolomonZelman

This is an example, read through it and ask if you have any questions.

7. ET365

So what happens the the dx?

8. SolomonZelman

So the $$\displaystyle \int_{a}^{b}$$ and $$\rm dx$$ are just indicating that you are integrating with respect to x, from x=a to x=b.

9. SolomonZelman

|dw:1443663482558:dw|

10. SolomonZelman

area between the x-axis and the function over the interval [a,b]

11. ET365

Ahh I gotcha, so theres another problem, but this one is (1+sqrt(49-x^2)dx. My problem here is figuring out the integral of the square root.

12. SolomonZelman

$$\large\color{black}{\displaystyle\int\limits_{~}^{~}\left(1+\sqrt{49-x^2}\right)~dx}$$

13. SolomonZelman

like this?

14. ET365

Yes, the period is betweek -7 and 0, but once I get to that portion its easy to figure out so I left out those numbers

15. SolomonZelman

Did you learn the *Trigonometric Substitution* method?

16. ET365

I was definitely supposed to learn that, but I've been drawing blanks from stress these past few days.

17. SolomonZelman

Do you have the limits of integration given?

18. ET365

Yes the (-7,0)

19. SolomonZelman

You can do reinman sums for now (with n=10 or something like this)..... or, you can read lamar tutorials about integration (they are very helpful), and then try your integral.

20. SolomonZelman

(I can go offline at any moment, I have to move on)

21. ET365

Ahh I understand thank you for your responses, they were extremely helpful!!

22. SolomonZelman

sure... whatever I can...

23. SolomonZelman

yw