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

• ibysaiyan
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.
ibysaiyan

## 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?

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?

y=lnx => x=e^y ?

Last edited:
still confused... but i thought just as -ln3 = 3^-1 = 1/3 ,same would happen to this but guess that's not the case ?

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).

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 .. =/

## 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.

### Similar threads

• Calculus and Beyond Homework Help
Replies
1
Views
809
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
6
Views
912
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
627
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
487
• Calculus and Beyond Homework Help
Replies
14
Views
513
• Calculus and Beyond Homework Help
Replies
6
Views
937
• Calculus and Beyond Homework Help
Replies
5
Views
1K