document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1085863>Question 1085863</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is  the centroid of the region bounded by the given curves.<BR>\n" );
document.write( "y = 6 sin(2x),    y = 6 cos(2x),    x = 0,    x = &#960;/8 \r<BR>\n" );
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document.write( "Thanks for your help .<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #699959 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=699959\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">Let<BR>\n" );
document.write( "f(x) = 6*sin(2x)<BR>\n" );
document.write( "g(x) = 6*cos(2x)<BR>\n" );
document.write( "Here is the graph of f(x) and g(x). They intersect at point A. The x coordinate of point A is pi/8 = 0.39 approximately<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/cMNMouL.png\"><BR>\n" );
document.write( "The region between the curves, from x = 0 to x = pi/8, is shown by the light blue shading<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/KTAfuK1.png\"><BR>\n" );
document.write( "Note: The fact that f(pi/8) = g(pi/8) = 3*sqrt(2) indicates that we would have a fully enclosed region without the need for the right boundary of x = pi/8, so it's a bit redundant. <BR>\n" );
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document.write( "The formulas we'll use can be found <a  rel=nofollow HREF=\"http://tutorial.math.lamar.edu/Classes/CalcII/CenterOfMass.aspx\">here</a>. Scroll down til you reach the \"Center of Mass Coordinates\" section. The formulas in the blue box below are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \bar{x} = \frac{1}{A}\int_{a}^{b}x*(f(x)-g(x))dx\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \bar{y} = \frac{1}{A}\int_{a}^{b}\frac{1}{2}\left((f(x))^2-(g(x))^2\right)dx\"><BR>\n" );
document.write( "which represent the coordinates of the centroid. The value of A is the area between the two curves, so,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A = \int_{a}^{b}(f(x)-g(x))dx\"><BR>\n" );
document.write( "Because the red g(x) curve is above the green f(x) curve all throughout the interval 0 < x < pi/8, this means that we must swap the locations and f(x) and g(x) when we subtract. So we should have these three formulas instead<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \bar{x} = \frac{1}{A}\int_{a}^{b}x*(g(x)-f(x))dx\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \bar{y} = \frac{1}{A}\int_{a}^{b}\frac{1}{2}\left((g(x))^2-(f(x))^2\right)dx\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A = \int_{a}^{b}(g(x)-f(x))dx\"><BR>\n" );
document.write( "This is to ensure A is positive and the centroid coordinates end up in the right spot.<BR>\n" );
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document.write( "We need to find the area A. Using numerical integration, I get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A = \int_{a}^{b}(g(x)-f(x))dx\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A = \int_{0}^{\pi/8}(6*\cos(2x)-6*\sin(2x))dx\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large A \approx 1.24264\"><BR>\n" );
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document.write( "Now use this to find the x coordinate of the centroid (xbar = <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \bar{x}\">)<BR>\n" );
document.write( "Again I'll use numerical integration to make things go quicker and more efficient<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \bar{x} = \frac{1}{A}\int_{a}^{b}x*(g(x)-f(x))dx\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \bar{x} \approx \frac{1}{1.24264}\int_{0}^{\pi/8}x*(6*\cos(2x)-6*\sin(2x))dx\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \bar{x} \approx 0.13365\"><BR>\n" );
document.write( "Do the same for the y coordinate of the centroid (ybar = <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \bar{y}\">)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \bar{y} = \frac{1}{A}\int_{a}^{b}\frac{1}{2}\left((g(x))^2-(f(x))^2\right)dx\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \bar{y} \approx \frac{1}{1.24264}\int_{0}^{\pi/8}\frac{1}{2}\left((6\cos(2x))^2-(6\sin(2x))^2\right)dx\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \bar{y} \approx 3.62132\"><BR>\n" );
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document.write( "We found that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(\bar{x}, \bar{y}\right) \approx \left(0.13365, 3.62132\right)\"><BR>\n" );
document.write( "Therefore, the centroid's location is approximately <font color=red>(0.13365, 3.62132)</font><BR>\n" );
document.write( "Here is an updated graph with the centroid point C added in<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/I9iIk5Q.png\"><BR>\n" );
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