Area of Region Bounded by Curve y=e^2x

In summary, to find the area of the region bounded by the curve y=e^2x, the x-axis, and the lines x=-ln3 and x=-ln2, use log laws and integration to solve for the area. Remember to be careful with the ln(1/6) term and to properly substitute in the bounds of integration.
  • #1
ibysaiyan
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0

Homework Statement


Find area of region bounded by curve with equation y=e^2x , the x-axis and the lines x=-ln3 and x=-ln2.


Homework Equations


log. law + integration


The Attempt at a Solution


well here is how i started this:
y=e^2x after integration
(1/2e^2x)
=>(1/2e^-(ln6)-(1/2^-(ln4) )
=>(1/2e^1/6) - (1/2e^1/4)
=>(1/2e^1/6)/ (1/2e^1/4)
=>(1/6(lne)1/2) / (1/4(lne)1/2)
(1/6X1/2) / (1/4X1/2)
=> is this correct?
 
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  • #2
ibysaiyan said:
y=e^2x after integration
(1/2e^2x)
=>(1/2e^-(ln6)-(1/2^-(ln4) )
=>(1/2e^1/6) - (1/2e^1/4)

e^(-ln6) is not equal to e^1/6; it's equal to e^ln(1/6). What happens to any number if you take its natural log and raise e to the result?
 
  • #3
y=lnx => x=e^y ?
 
Last edited:
  • #4
still confused... but i thought just as -ln3 = 3^-1 = 1/3 ,same would happen to this but guess that's not the case ?
 
  • #5
It is the case, but you wrote:

1/2e^-(ln6)

then, in the next step,

(1/2e^1/6)

That would mean -(ln6) is equal to 1/6, which it isn't. Instead, -(ln6) is equal to ln(1/6).
 
  • #6
ah k, now i get it yea.. 1/6 would only be equal to 6^-1 , erm.. sorry for being such a nuisance but now what to do with the ln(1/6)? , i can't ln both sides .. =/
 

Related to Area of Region Bounded by Curve y=e^2x

1. What is the formula for finding the area of a region bounded by the curve y=e^2x?

The formula for finding the area of a region bounded by the curve y=e^2x is ∫e^2x dx. This represents the integral of e^2x with respect to x, which can be solved using integration techniques.

2. How do you determine the limits of integration for finding the area?

The limits of integration for finding the area bounded by the curve y=e^2x depend on the specific region in question. Typically, the lower limit will be the x-value at which the curve intersects the x-axis, and the upper limit will be the x-value at which the curve intersects the y-axis. However, it is important to carefully analyze the graph of the curve to determine the appropriate limits.

3. Can the area of a region bounded by y=e^2x be negative?

No, the area of a region bounded by y=e^2x cannot be negative. The area under a curve is always a positive value, as it represents the space between the curve and the x-axis.

4. Is it necessary to use calculus to find the area of a region bounded by y=e^2x?

Yes, it is necessary to use calculus to find the area of a region bounded by y=e^2x. This is because the area under a curve is represented by an integral, which requires integration techniques to solve.

5. Can the area of a region bounded by y=e^2x be infinite?

No, the area of a region bounded by y=e^2x cannot be infinite. While the curve may extend infinitely in both the positive and negative directions, the area under the curve will always be a finite value.

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