Question

PLEASE HELP! MEDAL! @hero @mathmale @jim_thompson5910 @SolomonZelman

Answer 1

@iGreen @satellite73 @SolomonZelman @xapproachesinfinity

Answer 2

\(\color{#000000}{\displaystyle\int\limits_{4}^{5} (-35)~dx}\)

(The indefinite integral of -35 is just -35x)

\(\color{#000000}{\displaystyle\left. -35x {\color{white}{\Large |}}\right|_{4}^{5} =(-35\times 5)-(-35\times 4)=-175+140=-35}\)

Answer 3

This is an example, and I did it doing calculus, although you can use geometry too for constant functions

Answer 4

so, then it'll be -20, right?

Answer 5

No....

Answer 6

Why did you think that?

Answer 7

-20x?

Answer 8

yes, that would be the indefinite integral.

Answer 9

And in a case of your definite integral, your evaluating -20x from x=3 to x=7

Answer 10

Oh, okay. I don't completely understand how you get that though

Answer 11

-20x I mean

Answer 12

We can prove why derivative of ax (with respect to x) is just a, and then you know that integration and differentiation are operations that are reverses of each other.

Answer 13

would you like to prove this, or ... ?

Answer 14

So we're just finding the antiderivative?

Answer 15

Yes, the integral is antiderivative.

And in case when you have the limits of integration, then you are finding the derivative and then evaluating that at x=7 and x=3, and subtract.

Answer 16

Oh, ok. Integral notation always confuses me

Answer 17

The theory is:

So suppose that
\(\color{#000000}{\displaystyle\int\limits_{~}^{~} f(x)~dx=g(x)\color{grey}{+C}}\)

then,
\(\color{#000000}{\displaystyle\int\limits_{a}^{b} f(x)~dx=g(b)-g(a)}\)

Answer 18

is this making sense?

Answer 19

(as short and as precise as I can write it)

Answer 20

yes

Answer 21

Thank you. :)