Integral areas - Trig function

miss_b

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Joined
Sep 10, 2009
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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)!

y=(7.63)[sin(9.48x)]\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:

04(7.63)[sin(9.48x)]dx\displaystyle \int_{0}^{-4}(-7.63)[sin(9.48x)] dx

From what I have read thus far you take the (7.63)\displaystyle (-7.63) out infront of the 04\displaystyle \int_{0}^{-4} to become:
(7.63)04[sin(9.48x)]dx\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.
 
f(x) = 7.6340sin(9.48x)dx, Let u = 9.48x, then du9.48 = dx\displaystyle f(x) \ = \ -7.63\int_{-4}^{0}sin(9.48x)dx, \ Let \ u \ = \ 9.48x, \ then \ \frac{du}{9.48} \ = \ dx

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

= 7.639.48[1(.9757)] = .0196\displaystyle = \ \frac{-7.63}{9.48}[-1-(-.9757)] \ = \ .0196
 
:) :) :) 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)??
 
abf(x)dx = [F(x)]ab = F(b)F(a)\displaystyle \int_{a}^{b}f(x)dx \ = \ \bigg[F(x)\bigg]_{a}^{b} \ = \ F(b)-F(a)

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

ForExample: [cos(u)]37.920 = cos(0)[cos(37.92)] = cos(0)+cos(37.92)=1+.9757\displaystyle For Example: \ \bigg[-cos(u)\bigg]_{-37.92}^{0} \ = \ -cos(0)-[-cos(-37.92)] \ = \ -cos(0)+cos(37.92) = -1+.9757

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

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

For example: 7.6340sin(9.48x)dx = 7.6304sin(9.48x)dx\displaystyle For \ example: \ -7.63\int_{-4}^{0}sin(9.48x)dx \ = \ 7.63\int_{0}^{-4}sin(9.48x)dx
 
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