Sketch the region in the x y-plane defined by the inequalities x-2 y^2 ⩾0,1-x-|y| ⩾0 and find it

Sketch the region in the x y-plane defined by the...

Key Concepts

Intersection of Multiple Inequalities

In many problems, the region of interest is defined by the simultaneous satisfaction of several inequalities. Understanding how to find the overlap or intersection of these individual regions is crucial for determining the overall area or shape being analyzed.

Analysis of Quadratic Inequalities

Inequalities involving quadratic expressions define curved boundaries (such as parabolas) in the coordinate plane. Recognizing these curves and identifying their key features, like vertex and direction, is essential for accurately sketching the regions they delimit.

Area Calculation via Integration

Once a region is properly defined and sketched, calculus techniques such as integration are employed to precisely calculate its area. This often involves setting up appropriate definite integrals based on the boundaries of the region, and may exploit symmetry to simplify the computation.

Graphical Representation of Regions

This concept involves depicting sets of points in the coordinate plane that satisfy given inequalities. By translating algebraic inequalities into visual boundaries (such as curves or lines), one can determine and sketch the region of intersection. It is a fundamental concept in analytic geometry and aids in understanding the spatial relationship between different algebraic conditions.

Absolute Value Function and Inequalities

When inequalities involve the absolute value function, they typically produce regions that are symmetric with respect to the axis corresponding to the absolute value variable. Understanding how absolute value expressions work is key to correctly interpreting and sketching the bounds they impose on a region.

Transcript

00:01 We want to sketch the region defined by these two inequalities and then determine the area of that region.

00:07 Okay, so we're going to do a sketch in our xy plane.

00:11 As usual, we have here x and we have our y axes.

00:15 Okay.

00:17 So, with these inequalities, usually what we'd like to do is to solve for y as a function of x.

00:24 But however, because of this absolute value of y over here, we're actually going to solve for x in terms of.

00:32 Terms of y for both of these inequalities.

00:36 This is going to give us x is greater than or equal to 2y squared.

00:41 And in green we're going to have x is less than our equal to 1 minus the absolute value of y.

00:50 Ok, so now what we have here are two functions of y.

00:56 So when we plot these, we're going to kind of think of the graph as being sideways.

01:01 So let's plot the blue one first.

01:05 So let's take a look at x is equal to 2 y squared.

01:12 So thinking of x as, so we see that this is going to be a parabola, but it's going to be a sideways parabola, just by thinking of this with our head turned sideways.

01:25 And then what we're interested in is x being greater than or equal to 2 y squared.

01:30 So that's actually going to be the inside region here.

01:38 Now let's take a look at the green one.

01:40 We have x less than or equal to 1 minus y squared.

01:44 So first let's plot y equals 1 minus absolute y.

01:48 So we know that when y is 0, we have our point here at 1.

01:54 And then we're subtracting an absolute value, which means that as we go away from y equals 0, our x values will decrease.

02:02 So we are interested in x being less than or equal to 1 minus absolute y, which is going to give us this region over here.

02:12 So in fact we see that there is an enclosed region, or rather a region which overlaps between these two inequalities.

02:24 And let's color it in red.

02:28 So that's the area that we want to determine.

02:32 So it's going to be more instructive to do this as an integral of y.

02:39 So that we can integrate.

02:42 We see that this function is even or symmetric across this x -axis.

02:48 So if we just determine one of these areas, we can double it to determine the total area.

02:57 Okay, let's put this into equations.

03:02 Area is going to be equal to 2 times the area of, for example, let's take the top region...

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