## anonymous 4 years ago If the average rate of change of F on [1,3] is k, find ∫sin(t^2) dt [1,3] in terms of k

1. JamesJ

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

2. anonymous

yes i think this is a trap rule Q

3. JamesJ

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

4. anonymous

the first...the integral of

5. anonymous

F(t)=∫sin(t^2) dt

6. JamesJ

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 ...

7. JamesJ

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

8. anonymous

1/2 of the approximation of the integral?

9. JamesJ

No need to approximate, none at all.

10. anonymous

k

11. anonymous

they ask for it in terms of k

12. JamesJ

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

13. anonymous

1/2*k

14. JamesJ

No

15. anonymous

2K

16. JamesJ

Yes

17. anonymous

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

18. JamesJ

ok