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The area labeled $ B $ is three times the area labeled $ A $. Express $ b $ in terms of $ a $.

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$$b=\ln \left(3 e^{a}-2\right)$$

01:35

Frank Lin

00:44

Amrita Bhasin

02:22

Linda Hand

Calculus 1 / AB

Chapter 5

Integrals

Section 3

The Fundamental Theorem of Calculus

Integration

Ayman A.

March 25, 2021

If f is continuous and ?90f(x)dx=6, find ?30xf(x2)dx.

Oregon State University

Harvey Mudd College

University of Nottingham

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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in this problem, it is given that The area labeled B is three times the area of a. Express be in terms of a. Right? So this is the this is the graph given to us. Right? And this is the area. That is a given by eight. This is the area. Yeah, this is B. This is given by capital B. Area. Thank you. Mhm. Yeah. Right. So let me right here this is right. So area will be this uh this graph is nothing but this is the graph for each of the power X. Right? So we need to integrate from zero to A. That is the limit buttocks. Great. Simply here, zero to be for the grab video. 2nd seconds up. Right, So we are just calculating the area. Right? So we will be getting for area, area will be what this movie even be questioned it to dip our A minus what? Right. Similarly for area of the this is the you can say this is the question, nothing but the B minus one year. No. Oh you can say but it is given that area of triangle B is equally true three times of the idea. Right? So we are just putting this value. Right? This is very simple, isn't it? Yes. So we will put this value but we we are getting E to the power something. So to cancel out E. That is exponential form into, we need to put into normal forms of what we will do here, we will just take Ellen that islam. We know that log and seek answers each other. So this becomes B equals two. You can see Ellen 38 to the power a minus. So this is it. This. Mhm. In this way we got the area of be in terms of the So this is an answer. I hope you understood the concept. Thank you for watching.

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