Area problem

This one has me stumped, I dont even know where to start.
-Find the area of the triangular region in the first quadrant that is bounded above by the curve y=e^(2x), below by y=e^x, and on the right by the line x=ln(3).
Thanks for any help

Pyrrhus

Homework Helper
Try plotting the curves, and check definite integrals.

How would I go about checking for definate integrals? The book we use is not very descriptive on this topic. Thanks.

whozum

Plot the area, then find your integration limits. That's what he means by definite integral. Your final result will be a number, not an expression, representing the area in the shape.

HallsofIvy

Homework Helper
If you are doing a problem like this, then you certainly should be familiar with area given by integrals- that's the whole point of a problem like this. If you graph this, you will see that the two curves intersect at x= 0 and are bounded on the right by the vertical line x= 3. If you divide the area into thin rectangles, the height of each rectangle will be e3x- ex and the width will be "dx". Now, what integral is that?

mathwonk

Homework Helper
have you tried estimating it? i.e. can you show the area is less than 1,000?

or 10?

HallsofIvy

Homework Helper
Since this has been around for a couple of days now:

The graph of y= e3x is always above y= ex for x> 0. The two graphs cross at x= 0 and we are told that the are is bounded on the right by x= ln 3.

The area is $$\int_0^{ln 3}(e^{3x}- e^x)dx$$.

Can you do that integral?

geniusprahar_21

first draw the graph of all three fuctions. you'll get the picture. find the point of intersection of e^x, e^2x and x = ln3. this can be done easily...

one point is (0,1)...trivial solution
put x=ln3 in e^x and get y=3........(ln3,3)
put x=ln3 in e^2x and get y=9........(ln3,9)

now, for the next part you have to draw the graph.

integrate e^x limits 0 to 3 to get e^3 - 1
integrate e^2x limits 0 to 9 to get (e^9 - 1)/2

The area required is [(e^9 - 1)/2 - (e^3 - 1)]
note: area under graph of a function between a and b is its definite integral with a and b as lower and upper limits.

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