Question

Calculus
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Answer 1

You'll need knowledge the corollary to the FTC here.Do you understand the relationship between the area under a function and its integral?

Answer 2

yes

Answer 3

So what is another relationship between F(x) and f(x)?

Answer 4

or rather, what is the relationship after having applied the corollary to the FTC (sometimes mistaken as the FTC itself)?

Answer 5

I've never heard of corollary to the FTC before

Answer 6

Don't worry about it, you probably know it as the FTC itself

Answer 7

Graphically, what is F(x) given f(x)?

Answer 8

aren't they the same?

Answer 9

no....

Answer 10

\(\color{#000000 }{ \displaystyle {\rm F}(x)=\left.\int\limits_1^x f(t)~dt={\rm F}(t)\right|_1^x={\rm F}(x)-{\rm F}(1) }\)

\(\color{#000000 }{ \displaystyle ={\rm F}(x)-{\rm F}(1) }\)

Therefore, F(4) is going to be,

\(\color{#000000 }{ \displaystyle ={\rm F}(4)-{\rm F}(1) }\)

Answer 11

So, alternatively, that is the area between the f(x)
and the x-axis, on the interval \(x \in \{1,4\}\)

Answer 12

You can also see that the area on the interval [3 , 3.5],
and the area [3.5 , 4] have the same magnitude, EXCEPT
that the first one is negative and the other one is positive,
and thus they cancel themselves out.

Answer 13

so i calculate the area of the semi-circle, the rectangle on top, and the triangle?

Answer 14

Yes, exactly!
A rectangle 2 units by 1 unit.
Plus the area of the half-circle.

Answer 15

the triangles cancel out

Answer 16

yes.

Answer 17

so just semi-circle and rectangle?

Answer 18

Yes.

Answer 19

is it -2-(pi/2)?

Answer 20

yes, very good!

Answer 21

Thank you!

Answer 22

yw