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Use Riemann sums and the limit to find the area.Find the area of the region bounded by $f(x)=3 x+1$ and the $x$ -axis, between $x=0$ and 2.

$$8$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 5

Sigma Notation and Areas

Integrals

Missouri State University

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.

09:22

Use the method of Riemann …

01:52

Determine the area of the …

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03:31

Use Riemann sums and a lim…

03:06

mhm. Any time you're tasked with finding the area between a curve and they give you the curve and this problem as two over X and the X axis in the X axis, then what you want to do is the integral of that function DX, and they give you the bounds as well, they say from Mexico's 12 X equals three. And just make sure you go in order from least to greatest. So you get the right sign for your answer and we're ready to do the anti derivative of this. I don't know if you need a reminder that the derivative of natural log Becks is equal to one over X and because two is just a concept, you just move that, too in front, you know, it becomes two times one over X. If that helps you understand, the integral is to natural law defects. Because we're working backwards now. I'm in the habit of writing absolute value. You don't actually have to worry about absolute value in this problem because one and three are both positive. Uh, so, yeah, just plug in your balance to natural log of three minus, because that's how we evaluate to natural log of one. Now I expect my students to have it memorized. That natural log of one is equal to zero zero times. Anything is zero. And in this answer, minus zero is just that answer. So all of that simplifies to just to natural log of three, and that's a perfect answer now. I'm also well aware that another teacher might ask you to rewrite it. It's a law of logs, right? As natural log of three squared. If you move that experiment or natural log of nine just three times three or three squared is nine, but I don't know where to stop. This is good enough.

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