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

$ y = \tan x $ , $ y = \tan^2 x $ , $ 0 \le x \le \frac{\pi}{4} $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

Campbell University

University of Michigan - Ann Arbor

Idaho State University

Boston College

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.

13:47

Find the area of the regio…

04:49

0:00

07:52

05:12

03:59

02:47

Sketch the region bounded …

01:05

here we'd like to find the area of the region bounded by the curves. Ten x tan squared X for X between zero power for So here's a rough sketch of the graphs and let me explain why So, First of all, we know that can and tan squared or both zero on X zero and they're both won when excess power for now in between for exes between zero pyro for we have numbers. So we have tan and tan square. So we have zero less than our people two ten eggs less than equal to one on the interval. It's your own empire before, and I know that when you take a number between zero and one and you raise it to a higher power, their number itself gets smaller. So since the blue graph is a higher power of the same number ten, it's going to be smaller than the red graph. And so to confirm this, let's go to actual graphing calculator. So just this before the reddest hand, the bluest hand square and from zero to about power, for we see that the red graph is above the blue ref, so that tells us well First of all, we'LL have a formula for the area. So let's write that out. This area is the integral A to be so here. Zero power before of the absolute value of ten eggs Linus Tan square. And by our previous observation, we noticed that tan squared was below or less than or equal to ten eggs. This tells us that tan eggs minus ten square Dex is positive. And we can use this because if so, if this is positive, that means that the absolute value is just itself. So we can write this as in a girl zero power before ten eggs minus ten squared. Now let's go ahead and rewrite this zero power for CNX minus seek and squared X minus one. So we did here was use a path Agron identity to rewrite Tan Square a Sikh and squared minus one. We can evaluate all of these anti derivatives. The derivative of the tangent is natural log of absolute value of C can't. Here we have a minus tan X, and that becomes so we have a double minus their sort of plus X and rn points zero power for So it's going and plug in those end points. So is playing Pi over four first natural log seeking of Piper for his route, too. Since it's positive we can drop the absolute value there and then tangent apart before is one plus x. So plus power for and then when we plug in zero, we have natural log and it's seeking of zero is one tangent of zero zero and then plus zero. And we know that natural log of one zero So we could actually ignore this whole second term. And we're left over with our final answer. Ellen Route too minus one plus pirate for So that's our area, and that's our final answer.

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