Sketch the region enclosed by the given curves and find its area. y = (1)/(x) , y = x , y = (1)/

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

Key Concepts

Area Between Curves

Finding the area of the region enclosed by curves typically involves setting up definite integrals. This method computes the accumulated difference between the functions over the interval defined by their intersection points, representing the area contained between them.

Setting Up Limits of Integration

Once the curves and their points of intersection are identified, the next step is to determine the appropriate limits of integration. These limits represent the x-values (or y-values, if integrating with respect to y) over which the functions define the boundaries of the region of interest.

Curve Sketching

Curve sketching involves drawing the graph of functions to visually understand their behavior, shape, and relative position in the coordinate plane. It is a crucial step for identifying the region of interest and determining the boundaries defined by the given functions.

Intersection Points

Determining the intersection points of the curves is key in identifying where the boundaries meet. These points provide the limits for integration when calculating the area of the region enclosed by the curves, ensuring that the correct segment of the graphs is considered.

Transcript

00:01 We want to find the area of the enclosed region, closed by the functions y equals 1 over x, y equals x, and y equals 1 quarter times x.

00:15 So doing a quick sketch shows us that the region is this triangle over here.

00:21 So we do have to figure out what are these three points so that we know what will be our limits of integration.

00:28 So if we label these points 1, 2, and 3, let's take a look at 1 first.

00:37 So this point is very easy.

00:39 It's easy to see it's the origin because this is the intersection of the lines x and 1 quarter x.

00:46 Solving for this tells us x equals 0.

00:52 Let's take a look at the second point.

00:55 This is the intersection of the line x with the function 1 over x.

01:01 Solving for this gives us x squared equals 1, which gives us x is either plus 1 or minus 1.

01:08 But since we're only considering x greater than 0, this tells us that x is equal to 1 is the second point.

01:17 And now if we take a look at the third point, this is going to be the intersection of the line 1 quarter x with the function 1 over x.

01:29 So solving for this, we get x squared equals 4.

01:34 And then again, since x is positive, x is equal to 2.

01:38 So we can label these points over here with 0, x equals 0, x equals 1, and x equals 2.

01:48 So now we want to integrate to figure out the area.

01:53 So let's open up a new page.

01:56 Area is equal to, first we're integrating from 0 to 1...

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