1. ## integration

2 times the integral (from -3 to 3) of (4-x) (sqrt(9-x^2) ) dx.

Thank you very much.

2. Originally Posted by kittycat
2 times the integral (from -3 to 3) of (4-x) (sqrt(9-x^2) ) dx.

Thank you very much.
You question is:

Evaluate:

$\displaystyle \int_{-3}^3 (4-x) \sqrt{9-x^2}~dx$

RonL

3. Originally Posted by CaptainBlack
You question is:

Evaluate:

$\displaystyle \int_{-3}^3 (4-x) \sqrt{9-x^2}~dx$

RonL
$\displaystyle \int_{-3}^3 (4-x) \sqrt{9-x^2}~dx = \int_{-3}^3 4\sqrt{9-x^2}~dx - \int_{-3}^3 x \sqrt{9-x^2}~dx$

and these two integrals can be done by the usual book methods, the first by a trig substitution and the second
by looking at the derivative of $\displaystyle (9-x^2)^{3/2}$.

RonL

4. $\displaystyle \int_{-3}^3 (4-x) \sqrt{9-x^2}~dx$

Hint:
$\displaystyle = 4 \int_{-3}^3 \sqrt{9-x^2}~dx - \int_{-3}^3 x\sqrt{9-x^2}~dx$

You can do the second integral with a simple substitution. So let's look at the first term:
$\displaystyle \int_{-3}^3 \sqrt{9-x^2}~dx$

Let $\displaystyle x = 3y \implies dx = 3 dy$

Thus
$\displaystyle \int_{-3}^3 \sqrt{9-x^2}~dx = \int_{-1}^1 \sqrt{9-(3y)^2}~3dy$

$\displaystyle = 3\int_{-1}^1 \sqrt{9-9y^2}~dy$

$\displaystyle = 3\int_{-1}^1 3\sqrt{1-y^2}~dy$

$\displaystyle = 9\int_{-1}^1 \sqrt{1-y^2}~dy$

Does this integral look familiar?

-Dan

5. Or from the beggining you can use a simple trig. sub. defined by $\displaystyle x=3\sin u$

6. In general we have that

$\displaystyle \int\sqrt{a-x^2}\,dx=\frac12\left(x\sqrt{a-x^2}+a\arctan\frac x{\sqrt{a-x^2}}\right)+k$

7. Originally Posted by Krizalid
In general we have that

$\displaystyle \int\sqrt{a-x^2}\,dx=\frac12\left(x\sqrt{a-x^2}+a\arctan\frac x{\sqrt{a-x^2}}\right)+k$
I think you want to write $\displaystyle a^2$ (also mention that $\displaystyle a\not = 0$ , just to be safe, write $\displaystyle a>0$).

8. Originally Posted by ThePerfectHacker
I think you want to write $\displaystyle a^2$ (also mention that $\displaystyle a\not = 0$ , just to be safe, write $\displaystyle a>0$).
And mention that a is a real number, and...

-Dan

9. Hi Dan,

how do I end this one?

9 times the integral from -1 to 1 of sqrt(1-y^2) dy - your last step! Really don't know how

Also,
Can anyone remind me how to do the trig substitution? I forgot it ! Please show me the steps . Thanks.

10. Set $\displaystyle y=\sin u\implies dy=\cos u\,du$

Now when you have all integrated you use the first substitution to turn the trig. expressions into $\displaystyle y$ expressions.

11. hi Krizalid,

Please show me more ...! really forgot ! If possible , please show me the steps how to show the first integral by trig substitution. thank you very much.

12. Change of variables according to $\displaystyle y=\sin u$ implies

\displaystyle \begin{aligned} \int {\sqrt {1 - y^2 } \,dy} &= \int {\sqrt {1 - \sin ^2 u} \cos u\,du}\\ &= \int {\cos u\cos u\,du}\\ &= \int {\cos ^2 u\,du} \end{aligned}

Which is easy to take.