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Determine a Riemann sum that would determine the area of the region bounded by $f(x)=e^{x}$ and the $x$ -axis, between $x=0$ and 1 (do not evaluate this limit).

$$\lim _{n \rightarrow \infty} \sum_{i=1}^{n} e^{i \ln 1} \frac{1}{n}$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 5

Sigma Notation and Areas

Integrals

Missouri State University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

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.

02:05

Write a Riemann sum and th…

03:26

03:49

04:59

Evaluate the integral by c…

the question gives us this rectangle and asks us to use a remand some to evaluate the area. So to start we're just going to imagine that this rectangle is on a coordinate plane with this being the X. Axis, this is the Y. Axis and this as the origin. So now we could imagine that this rectangle sits between zero and five on the X axis and is bounded by the function Y equals three. And so now we can put this into a remand some which is going to look like the limit as N approaches infinity Of the some from I equals one to end. And now we want to find the equation giving us the area of this rectangle we're going to use and to remain some. So our leg times with is the equation for the area of rectangles or length is going to be Just equal to three and times our wit, which is just our delta X. Down here. So we have three times delta X. And now this is our remain some here and now we can put this into an integral, Which is going to take the form the definite integral from 0 to 5 of three times delta x. And now we just need the anti derivative of three, Which is going to be three x from 50 And now plugging into solve, we get three times five minus three times zero, Which is equal to 15. And this is the area of the shaded region.

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