Sketch the region enclosed by the given curves and find its...

Key Concepts

Graphing Functions

Graphing functions involves plotting the given curves to understand their shapes and relative positions. This is essential for visualizing the region of interest and identifying the boundaries that enclose the region whose area is to be calculated.

Piecewise Functions

A piecewise function, such as the absolute value function, is defined by different expressions over different parts of its domain. Recognizing this helps in correctly sketching the function and understanding how it behaves over different intervals.

Intersection of Curves

Determining the points where curves intersect is crucial because these points define the limits of the region being analyzed. This involves solving equations set equal to each other, which provides the necessary bounds for subsequent calculations, such as integration.

Definite Integration for Area

Definite integration is used to evaluate the area between curves by integrating the difference between the upper and lower function values over the interval defined by the intersection points. This method is a core technique in calculus for calculating areas of irregular regions.

Symmetry in Functions

Recognizing symmetry in functions can simplify the integration process. When a region is symmetric about an axis, the area calculation may be reduced by computing the area of one symmetric part and then multiplying by the appropriate factor to obtain the total area.

Transcript

00:01 Okay, so if you were to graph this, i believe it's 2 over 1 minus, sorry, 1 plus x squared.

00:14 And then the absolute value function looks something like this.

00:19 I'm going to put arrowheads on this.

00:21 So this is the absolute value of x.

00:23 So what i would do, because these are even functions, and i hope that makes sense to you, is i would just find this integral and then double your answer.

00:33 So, well, how do i find that integral is it's the integral from zero because that's the x.

00:41 And where these two intersect is that x equals 1 because if you plug in 1 for x squared is 1 plus 1 is 2, 2 divided by 2 is 2 is 1 and the absolute value of 1 is still 1.

00:56 So the upper function is that 2 over 1 plus x squared minus this value.

01:04 Is just, you know, x at this point because of, you know, we drop the absolute value because we're only looking at positive values for x.

01:17 And so then what i need to do, as i mentioned earlier, is double that answer.

01:21 So now we can go ahead and take the integral of that.

01:24 So that two is still in front.

01:26 So i hope you've already learned this.

01:29 Otherwise, it would take me forever to teach that.

01:31 That antiderivative is the inverse tangent of x.

01:35 You can just google what is the integral of 1 over 1 plus x squared if you didn't know that.

01:41 Now this 2 is just a constant so you can bring it in front...