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Find the area of the region bounded by the given curves.

$ y = x^2 \ln x $ , $ y = 4 \ln x $

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$\frac{16}{3} \ln (2)-\frac{29}{9}$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 1

Integration by Parts

Integration Techniques

Missouri State University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

05:52

Find the area of the regio…

02:01

the problem is find the area of the region only by the given curves here. Why is they go to High Square l N X and why is they go to four times l n X? First we lied. These two functions uh, echo and find intersection Point. Let's say this is X choir minus or times Helen likes is equal to zero. This X is greater than zero. We have X is equal to tool or Alan. Tax is 0 to 0. So this actually why Mhm So the interval is from 1 to 2 and what acts is between one and two. Four times L and X is great as on I squared times our next since I rear bounded by given curves is integral from 1 to 2 or times are impacts minus X choir times out next thanks, or this is equal to the integral from 1 to 2 four minus X choir. Alan Nex Next. Now we can use integration by parts to southeast. Integral former is integral of U K. Prime Detox is legal to you'll be minus the integral of your prime rejects. Now we can let you is the co two warm minus X choir. And we now when u is equal to Highland Max and we promise to four minus x squared, then well, you promise people wanna racks V a secretary or acts minus what third tax to three is power, then this integral. It's a cultural new times we So this is or X minus one third x two threes power times l and X from 12 to minus interior from 1 to 2. You prime time, sweetie. So this is more minus my dude X square. Mm x such as is you, can't you? So for the first term, we plug in two and 12 blocks than behalf. This is eight minus it over. Three times Ellen Chew and minus zero and minus. And to grow this oneness war axe minus one. Number nine x two threes. Power from one to this is true. 16/3 hour in two minus This one who is eight minus 8/9 minus four, minus one over I. So this is the result

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