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Determine the area of the indicated region.Region bounded by $f(x)=3 e^{x}$ and the $x$ -axis, between $x=0$ and 2.

$$3\left(e^{2}-1\right)$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 6

The Definite Integral

Integrals

Oregon State University

Baylor University

Boston College

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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Here's another area problem between the curve, which they give you as three e to the X, um, and the X axis. So the area between the curb and the X axis that's important just for this class right now, you just have to recognize that That's saying, Hey, we're doing the integral and then they give you the bounds and you want to go from least to greatest. Um So where they give you X equals zero and X equals two. Then we can set up this equation from 0 to 2. Smallest, greatest three e to the X DX. Um and the only thing to really think about it is if I asked you for the derivative of some constant times either the X. I hope you would all say that that answer is able to see either the X so going backwards for the anti directive or the integral should be the same thing. Three e to the X from 0 to 2. Now, I would probably factor out that three that is a rule of of constant. You can just move that constant in front. So I'm just thinking of this problem. But some people like to distribute the three in. It doesn't really matter. But if I were doing this and plugging in my bounds, I would just leave it like this because some teachers like to factor out that that constant. Anyway, um, so then I would also expect my students to know that each of the zero power is equal to one. So I wouldn't let my students leave their answer like this. I would go ahead and rewrite. It is three times a quantity B squared, minus one. Um, and that's a good answer. You could get by with distributing that three. And like I mentioned earlier, um, and this is a good answer as well. So either answer Circled in green is good.

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