Finding the Area of a Triangular Region in the First Quadrant

[shadow5449]

This one has me stumped, I don't 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]

Try plotting the curves, and check definite integrals.

 

[shadow5449]

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]

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 e^3x- e^x and the width will be "dx". Now, what integral is that?

 

[mathwonk]

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

or 10?

 

[HallsofIvy]

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

The graph of y= e^3x is always above y= e^x 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.