anonymous 5 years ago hey can someone please help me find the integral of sqrt of y^2-49. i know i have to use trig substitution but i need help with actually solving it

1. myininaya

$\int\limits_{}^{}\sqrt{y^2-49}dy$ let $\frac{y}{7}=\tan(\theta)$ $\frac{dy}{7}=\sec^2\theta d \theta$ $\int\limits_{}^{}\sqrt{(7\tan \theta)^2-49}7\sec^2\theta d \theta$

2. anonymous

isnt it better to use 7 sin theta as a sunstitution

3. myininaya

$7\int\limits_{}^{}\sqrt{49}\sqrt{\tan^2\theta-1}\sec^2(\theta)d \theta$

4. myininaya

$7*7\int\limits_{}^{}\sec \theta*\sec^2(\theta) d \theta$

5. myininaya

$49\int\limits_{}^{}\sec^3(\theta) d \theta$

6. myininaya

oops i made the wrong substitution

7. anonymous

yah lol

8. myininaya

tan^2x+1=sec^2x not tan^2x-1=sec^2x

9. anonymous

could you do it using sin theta as the substitution if it not too much trouble

10. anonymous

this is the part im stuck if it helps

11. myininaya

you could also use sintheta=y/7

12. anonymous

i got 49 cos^2 theta

13. myininaya

i mean costhetha=y/7

14. anonymous

then i knew i i got the integral of cos^2 theta which came out to ve 1/2 theta -1/4 cos 2 theta

15. myininaya

$\sin \theta=\frac{y}{7}$ $\cos(\theta)d \theta=\frac{dy}{7}$ $(\sin \theta)^2=(\frac{y}{7})^2$ $49\sin^2(\theta)=y^2$ $\int\limits_{}^{}\sqrt{49\sin^2(\theta)-49}*7\cos(\theta)d \theta$ $7\int\limits_{}^{}\sqrt{49}*\sqrt{\sin^2(\theta)-1}*\cos(\theta)d \theta=49\int\limits_{}^{}\cos^2\theta d \theta$

16. myininaya

remember cos^2(theta)=1/2*(1+cos(2theta))

17. myininaya

$49*\frac{1}{2}\int\limits_{}^{}(1+\cos(2\theta)) d \theta=\frac{49}{2}*(\theta+\frac{1}{2}\sin (2\theta))+C$

18. myininaya

but remember sin(2theta)=2sin(theta)cos(theta)

19. myininaya

$\frac{49}{2} \theta+\frac{49}{2} \sin(\theta)\cos(\theta)+C$

20. myininaya

we need this in terms of x though we let sintheta=y/7 so costhetha=sqrt(49-y^2)/7

21. myininaya

also since sintheta=y/7 then theta=arcsin(y/7)

22. myininaya

meant in terms of y lol

23. anonymous

hey i forgot to give you the limits bc i am having trouble plugging it its 0 to 7

24. myininaya

$\frac{49}{2}\sin^{-1} (y)+\frac{49}{2}\frac{y}{7}\frac{\sqrt{49-y^2}}{7}+C$

25. myininaya

$\frac{49}{2}\sin^{-1} (y)+\frac{y}{2}\sqrt{49-y^2}+C$

26. myininaya

you are having trouble plugging in?

27. myininaya

wherever you see a y plug in your upper limit then minus wherever you see a y plug in your lower limit

28. myininaya

$(\frac{49}{2}\sin^{-1}(7)+\frac{7}{2}\sqrt{49-7^2})-(\frac{49}{2}\sin^{-1}(0)+\frac{0}{2}\sqrt{49-0^2})$

29. myininaya

$\frac{49}{2}\sin^{-1}(7)$

30. anonymous

ty :)