anonymous 5 years ago how do you find the integral of the following: integral sqrt 9-h^2 limits 0 to 3

1. anonymous

Are you learning about trig substitutions or polar coordinates?

2. anonymous

hey i have a test and i am review for it, its trig sunsitution

3. anonymous

substitute = 3 sin x

4. anonymous

Since in the limit of integral,0<h<3, so u can substitute h=3sin(a) since sin is btw 0 and 1, then (9-h^2) becomes 9cos^(a) take sqrt=3|cos(a)| also dh=d(3sin(a))=3cos(a)da thus and change limits h=0=3sin(a)=>a=0 h=3=3sin(a)=>a=pi/2 thus integral becomes limit 0 to 1 9cos^2(a)da put cos^2(a)=(2cos^(2a)-1)/2 then integrate

5. anonymous

Okay, since we have sqrt(a^2 - u^2) form, we have to use a sin t substitution. Let u = asint $h = 3\sin \theta$ You should find that sine of theta is equal to h / 3. Use SOHCAHTOA to draw a right triangle. One leg should be h and the hypotenuse should be 3. Find the other leg using the pythagorean theorem to get sqrt(9 - h^2). Take the cosine of the angle $\cos \theta=\sqrt {9-h^2}/3$ Multiply by 3 on both sides to get cosine(theta)/3 to use as a substitution. Substitute that for sqrt(9 - h^2) in the integral. Now go back to the h = 3sin(theta) equation and take the derivative to get dh = 3cos(theta). Substitute 3cos(theta) for dh to get the integral: $\int\limits_{\theta(0)}^{\theta(3)}\cos \theta /3*3\cos \theta d \theta$ $\int\limits_{\theta (0)}^{\theta (3)}\cos ^2 \theta d \theta$ You'll then have to integrate that and back-substitute.

6. anonymous

you can also draw a circle with radius 9 and conclude that the area is just one-fourth of the circle...

7. anonymous

8. anonymous

okay thank you for the explanation but this question but this question is related to another problem u think u can help me

9. anonymous

ok

10. anonymous

can u please click on the link to see the picture i have to find the area by writing the definite integral and evaluate it http://www.twiddla.com/504415

11. anonymous

What area do you need? The area between the two chords?

12. anonymous

no thats the strip and u have to use the strip to find the area of the entire circle

13. anonymous

you would have to first write riemann sum then the definite integral