Integral areas - Trig function

[miss_b]

Hey guys!

I'm trying to solve and I'm not getting anywhere (both my text books and the net aren't helping me very much tonight)!

\(\displaystyle y = (-7.63)[sin(9.48x)]\) x=-4 and x=0

Answer:
Integral Area = 0.0196 (to four decimals)

We have only just started learning Integrals (and as I didn't finish high school over 10 years ago) I'm really struggling with this topic. I understand that integration is basically the oppersite of differentiation. SO, the integral of sin is -cos (I think).

Here is what I think:

\(\displaystyle \int_{0}^{-4}(-7.63)[sin(9.48x)] dx\)

From what I have read thus far you take the \(\displaystyle (-7.63)\) out infront of the \(\displaystyle \int_{0}^{-4}\) to become:
\(\displaystyle (-7.63)\int_{0}^{-4}[sin(9.48x)] dx\) ** I just pulled that from an example I read here on this site that one of the admins did for another member**

Thanks for your time.

 

[BigGlenntheHeavy]

\(\displaystyle f(x) \ = \ -7.63\int_{-4}^{0}sin(9.48x)dx, \ Let \ u \ = \ 9.48x, \ then \ \frac{du}{9.48} \ = \ dx\)

\(\displaystyle Ergo, \ \frac{-7.63}{9.48}\int_{-37.92}^{0}sin(u)du \ = \ \frac{-7.63}{9.48}\bigg[-cos(u)}\bigg]_{-37.92}^{0}\)

\(\displaystyle = \ \frac{-7.63}{9.48}[-1-(-.9757)] \ = \ .0196\)

 

[miss_b]

You guys are so awesome!! Thanks again Glenn

Just the one bit I am now confused about in your reply. In the last step, where does the (-1) come from? Is it the derivative of x in the original equation, but it is a negative cause of the -cos(u)??

 

[BigGlenntheHeavy]

\(\displaystyle \int_{a}^{b}f(x)dx \ = \ \bigg[F(x)\bigg]_{a}^{b} \ = \ F(b)-F(a)\)

\(\displaystyle FTC, \ just \ follow \ the \ procedure.\)

\(\displaystyle For Example: \ \bigg[-cos(u)\bigg]_{-37.92}^{0} \ = \ -cos(0)-[-cos(-37.92)] \ = \ -cos(0)+cos(37.92) = -1+.9757\)

\(\displaystyle Note: \ \int sin(\theta)d\theta = -cos(\theta)+C \ and \ cos(-\theta) \ = \ cos(\theta), \ even \ function.\)

\(\displaystyle Addendum: \ -c\int_ {a}^{b}sin(\theta)d\theta \ = \ c\int_{b}^{a}sin(\theta)d\theta\)

\(\displaystyle For \ example: \ -7.63\int_{-4}^{0}sin(9.48x)dx \ = \ 7.63\int_{0}^{-4}sin(9.48x)dx\)

 

[miss_b]

Thanks again BigGlenn