area bounded by curves

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October 21st 2009, 03:45 AM
zerocool18

area bounded by curves

hello , ive been stuck on this problem all night now. my heads starting to hurt lol. anyway im supposed to determine the area of the region by the graphs of the equations y=sin^2(pix) , y=0 , x=0 , x=1.

heres a link to what the equation looks like , just in case http://www.wolframalpha.com/input/?i=integrate+sin^2%28pi*x%29+dx+from+x%3D0+to+1

now, the answer i get is 3/4, which i think is wrong. i replaced sin^2(pix) with 1-cos^(pix).
October 21st 2009, 04:39 AM
calum

You need to use the double angle formula: $cos(2x) = 1-2sin^2(x)$

Hence $sin^2(\pi x) = \frac{1}{2} - \frac{1}{2}cos(2 \pi x)$
October 21st 2009, 04:57 AM
tonio

Quote:

Originally Posted by zerocool18

hello , ive been stuck on this problem all night now. my heads starting to hurt lol. anyway im supposed to determine the area of the region by the graphs of the equations y=sin^2(pix) , y=0 , x=0 , x=1.

heres a link to what the equation looks like , just in case http://www.wolframalpha.com/input/?i=integrate+sin^2%28pi*x%29+dx+from+x%3D0+to+1

now, the answer i get is 3/4, which i think is wrong. i replaced sin^2(pix) with 1-cos^(pix).

$\mbox{You can check that}\,\int \sin^2x\,dx\,=\frac{x-\sin x\cos x}{2}+C \,$ $\,\mbox{, with }\,C \,\mbox{a constant, so a simple substitution will give you}$

$\int_0^1 \sin^2\pi x\,dx=$ $\frac{1}{\pi}\int_0^\pi\sin^2x\,dx$

Tonio
October 21st 2009, 05:16 AM
zerocool18

i think i got it , just one more question. Im supposed to use A=f(x)-g(x) to find the area. since the graph doesnt intersect anything, my equation should be the integral from 0 to 1 of sin^2(pix)-0 right? so i just have to integrate the sin^2(pix)?
October 21st 2009, 05:53 AM
tonio

Quote:

Originally Posted by zerocool18

i think i got it , just one more question. Im supposed to use A=f(x)-g(x) to find the area. since the graph doesnt intersect anything, my equation should be the integral from 0 to 1 of sin^2(pix)-0 right? so i just have to integrate the sin^2(pix)?

Yes

Tonio