Problem

Sketch the region enclosed by the curves and find…

01:23

Problem 13 Medium Difficulty

Sketch the region enclosed by the curves and find its area.
$$
y=e^{x}, y=e^{2 x}, x=0, x=\ln 2
$$

Answer

$$
A=1 / 2
$$

Related Courses

Calculus 1 / AB

Calculus 2 / BC

Calculus Early Transcendentals

Chapter 6

APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING

Section 1

Area Between Two Curves

Discussion

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Video Transcript

So I'm going to give a very lame graph here because an exponential function. Either the ex looks something like this is the ordered pair. 01 by the way, Actually, doesn't matter. But, um so this would be eating my ex now e to the X squared grows exponentially faster. I'm like this each of the two X um just think if you're doubling your exponents, it's gonna, well, horizontal squeeze if you will, um, and they cross the y axis at the same point. And the reason why I say it doesn't matter is then they also tell you that they want one bound to be X equals zero and the other bound to be X equals. So these air vertical lines, um, natural log of to So what we need to do is the integral from zero the lower bound, which he told you to the upper bound of natural, uh, give to of the upper function Eat of the two x minus e to the X DX. Now I would not use u substitution if you want my opinion as I would just think. OK, well, if it's e to the two x well, if I did the derivative of each of the two X I would get either of the two x times to cause a chain rule. Well, how do you fix the times? Two is sticking a one half in front and can double check that. This is correct, cause if you do each other to actually get you to the two x Times two, which cuts out which cancels out with this one half. So we're good there, minus the drift of Evita, the exes, you the x from zero to natural log of to. So as we plug things in, um, you get one half eat the to natural log of to minus E to the natural log of to minus one half E to the zero minus B 20 And you might say, Well, two times, you're still zero. I just wrote that. Um, Now I would do a lot of work just to make sure I'm doing this. Correct that, Cheech colors change. Okay, I would actually show the work of moving this to as an excellent that your law of logs, and then you could cancel this out. But this is really two squared, which is four times. One half. That's how you get to is that first piece and then here. Um, either the natural log cancels out kind of the same way. Is that? And so you just left with two. Now, either the zero equals one. So you're ending up with one half minus heated. The zero is one. So, as you're looking at this end of two months to a zero minus one half, minus one is negative one half Well, zero plus one half equals one half, which is your answer.

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