Question

If the average rate of change of F on [1,3] is k, find
∫sin(t^2) dt [1,3] in terms of k

Answer 1

...you mean where
\[ F(t) = \int \sin(t^2) \ dt \]

Answer 2

yes
i think this is a trap rule Q

Answer 3

or where \( F(t) = \sin(t^2) \), or what? What's your definition of F?

Answer 4

the first...the integral of

Answer 5

F(t)=∫sin(t^2) dt

Answer 6

Well, by the Fundamental Theorem of Calculus, the rate of change of F, is the derivative dF/dt is given by
\[ dF/dt = \sin(t^2) \]

Now integrals find averages of things. The average of a function \( f(t) \) over an interval [a,b] is given by
\[ \frac{1}{b-a} \int_a^b f(t) \ dt \]

You're told that
\[ \frac{1}{3-1} \int_1^3 \sin(t^2) \ dt = k \]
Hence ...

Answer 7

Hence what must
\[ \int_1^3 \sin(t^2) \ dt \]
be equal to?

Answer 8

1/2 of the approximation of the integral?

Answer 9

No need to approximate, none at all.

Answer 10

k

Answer 11

they ask for it in terms of k

Answer 12

Yes ... read again the last equations I wrote up there for you.

Answer 13

1/2*k

Answer 14

No

Answer 15

2K

Answer 16

Yes

Answer 17

ok....thank you so much for walking me thru it

Answer 18

ok