Math Help - evaluate the integral

1. evaluate the integral

$\int_{-5}^5 (8-|x|) dx$
the answer is 55 but I am hving throuble getting to the solution.

2. Re: evaluate the integral

Originally Posted by delgeezee
$\int_{-5}^5 (8-|x|) dx$
the answer is 55 but I am hving throuble getting to the solution.
Since you have a even function:

$\int_{-5}^5 (8-|x|) dx=2\int_{0}^5 (8-|x|) dx=2\int_{0}^5 (8-x) dx$

CB

3. Re: evaluate the integral

Originally Posted by delgeezee
$\int_{-5}^5 (8-|x|) dx$
the answer is 55 but I am hving throuble getting to the solution.
$|x| =\begin{cases} -x, & \text{if }x \leq 0 \\x, & \text{if}x\geq 0\end{cases}$

$I=\int\limits_{-5}^{0} (8+x)\, dx+ \int\limits_{0}^{5}(8-x)\, dx$

4. Re: evaluate the integral

Sorry! I can solve it using geometry, but I still need a little more help.

The interval length I had before was 10/n, would it now change to 5/n ??

$\frac{5}{n} [\sum\limits_{k=1}^{n} (8+x) + \sum\limits_{k=1}^{n} (8-x) ]$

Using reimann's right sum $(a+k* \Delta )$

does it matter which reimann sum I use?

5. Re: evaluate the integral

Why are you doing that? Are you required to reduce it to a Riemann's sum first? If so you should have told us that to start with.

6. Re: evaluate the integral

According to Captain Black's post, your area is $2\int_{0}^{5}(8-x)dx$.

Now, recall that $\int_{a}^{b}f(x)dx=\lim_{n \to \infty}\sum_{i=1}^{n}f(x_i)\Delta x$ where $\Delta x=\frac{b-a}{n}$ and $x_i=a+i \Delta x$.

In this case

$a=0$
$b=5$
$\Delta x = \frac{5}{n}$
$x_i=0+i\Delta x =\frac{5i}{n}$
$f(x_i)=8-\frac{5i}{n}$

required area = $2 \times \lim_{n \to \infty}\sum_{i=1}^{n}( 8-\frac{5i}{n})\frac{5}{n}$

Evaluating this, you will get the value of the definite integral.

7. Re: evaluate the integral

Originally Posted by delgeezee
Sorry! I can solve it using geometry, but I still need a little more help.

The interval length I had before was 10/n, would it now change to 5/n ??

$\frac{5}{n} [\sum\limits_{k=1}^{n} (8+x) + \sum\limits_{k=1}^{n} (8-x) ]$

Using reimann's right sum $(a+k* \Delta )$

does it matter which reimann sum I use?
Post the full question.

CB

8. Re: evaluate the integral

Ti-89 says 55 you got the correct answer.

9. Re: evaluate the integral

Originally Posted by CalBear12
Ti-89 says 55 you got the correct answer.
Yes, 55 is the correct answer, but the point is that delgeeze never got the answer.

10. Re: evaluate the integral

Originally Posted by sbhatnagar
Yes, 55 is the correct answer, but the point is that delgeeze never got the answer.
I'm afraid most of CalBear12's posts partake of the nature of the one you are commenting upon. I'm hopping he will learn to read the original and the other posts in a thread more carefully before shooting-form-the-hip before he earns a weeks ban due to accumulated infractions.

CB