Sketch the region bounded by the given curves and find the area of the region. y=1 / x, y=1 /

Sketch the region bounded by the given curves and find the...

Key Concepts

Graphical Analysis of Functions

Understanding the behavior and shape of the functions involved is crucial. This includes sketching the curves on a coordinate plane by noting key features such as asymptotes, intercepts, and the general trend of the function. Graphical analysis helps in visualizing how the curves interact with each other and aids in determining the region whose area needs to be calculated.

Finding Intersection Points

Determining where the curves intersect is necessary to define the limits of the region. This involves solving the equations of the curves simultaneously. The intersection points serve as integral bounds or help indicate where one curve transitions relative to the other in the area computation.

Defining the Bounded Region

Identifying and outlining the region enclosed by the curves is an important step. This involves understanding which curve lies above or below another in the interval of interest and recognizing any vertical or horizontal boundaries. A clear definition of the bounded region is essential to correctly set up the definite integral for area calculation.

Definite Integration to Compute Area

Once the bounded region is defined, the area is typically calculated using definite integrals by integrating the difference between the functions across the given interval. This requires properly setting up the integral with the correct limits and integrands based on the relative positions of the curves, ensuring that the integrals represent the area between these curves accurately.

Transcript

00:03 In this question, we're asked to find the area in between three curves, which is the region that is shaded in blue.

00:17 Now, one thing to notice is that at specific points, the regions change which curve is the top curve and which curve is the bottom curve.

00:28 So how do we deal with that? we just separate them into two integrals, and then we'll just use them.

00:43 To find the intersection points.

00:46 Now, what are our intersection points? to start, let's begin with the two linear functions.

01:01 We know that for x to be equal to x over 8, pretty obvious, x has to be 0.

01:15 Now focusing with the two linear, with a two with a line and the inverse function, we're going to have that x has to equal 1.

01:26 Again, it's fairly self -evident.

01:32 And to deal with the bottom one, we're going to have to have the x equals 2.

01:45 So why did i choose these three points? you can verify this on desmos or any other graphing calculator of your choice, but you'll notice that these are the intersection points.

01:56 And if i was to plug in any other values of x, then none of these intersection points would be, well, intersection points.

02:06 So now that we know this, we remember that the area in between two curves on some interval is given by f or by a top function minus a bottom function.

02:27 And as you can see, since we have two different top functions, so we have to separate this into two integrals.

02:35 So this will be an integral from 0 to 1.

02:41 From 0 to 1, we only care about x and x over 8...

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