document.write( "Question 574997: Sketch the region enclosed by y = e^{3x}, y=e^{7x} and x=1.\r
\n" ); document.write( "\n" ); document.write( "Decide whether to integrate with respect to x or y, and then find the area of the region. \n" ); document.write( "

Algebra.Com's Answer #369445 by KMST(5328) $\"\"$ $\"About$

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At x=0, y=1 for both functions, so they intersect at (0,1). The region would have vertical line x=1 and the two functions as boundaries. The region has the graph of $\"y=e%5E%287x%29\"$ as the upper boundary for y, and the graph of $\"y=e%5E%283x%29\"$ as its lower boundary throughout the [0,1] interval.
\n" ); document.write( "Integrating with respect to x would simply mean
\n" ); document.write( " $\"int%28+%28e%5E%287x%29-e%5E%283x%29%29%2C+dx%2C+0%2C+1+%29\"$
\n" ); document.write( "If we try to integrate with respect to y, we need to do some calculations first:
\n" ); document.write( " $\"y=e%5E%283x%29\"$ and $\"x=1\"$ intersect at (1, $\"e%5E3\"$ ), and
\n" ); document.write( " $\"y=e%5E%287x%29\"$ and $\"x=1\"$ intersect at (1, $\"e%5E7\"$ )
\n" ); document.write( "The y values for the region range between 1 and $\"e%5E7\"$
\n" ); document.write( "The inverse of $\"y=e%5E%287x%29\"$ is $\"x=ln%28y%29%2F7\"$ , which is the lower boundary for x values in the region.
\n" ); document.write( "On the other hand, the upper boundary for x values is $\"x=ln%28y%29%2F3\"$ between 1 and $\"e%5E3\"$ ), and x=1 between $\"e%5E3\"$ ) and $\"e%5E7\"$ ).
\n" ); document.write( "So after all those calculations, we would end up with two integrals.
\n" ); document.write( " $\"int%28+%28e%5E%287x%29-e%5E%283x%29%29%2C+dx%29=e%5E%287x%29%2F7-e%5E%283x%29%2F3\"$ , so
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= approx. 150.157 \n" ); document.write( "

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