Determine the shaded areas shown in Fig. 7 below:

Name: Problem 3, Chapter 8: Engineering Mathematics Through Applications
Uploaded: 2021-08-12T00:51:14-08:00
Duration: 3 min 12 s
Channel: Gregory Higby
Description: Determine the shaded areas shown in Fig. 7 below:

Key Concepts

Geometric Decomposition

Geometric decomposition involves breaking a complex shape into simpler components whose areas can be calculated easily. After calculating these simple areas, they are either added together or one is subtracted from another to arrive at the total shaded area.

Subtraction Principle in Area

The subtraction principle is used when the area of an unwanted region is removed from a greater area to obtain the area of the desired shaded part. This method is essential when the overall figure encompasses both shaded and unshaded areas, necessitating the removal of one part to calculate the other.

Area Calculation

Area calculation involves determining the measure of a two-dimensional region enclosed by boundaries. This is done using standard formulas for standard shapes or, in more complex cases, by applying calculus techniques such as integration to compute the area under curves.

Composite Figures

Composite figures are shapes that can be divided into simpler, known geometric figures. Understanding how to decompose a complex figure into basic shapes—such as rectangles, triangles, circles, etc.—allows for the computation of its total area by summing or subtracting the areas of these simpler components.

Shaded Regions

Shaded regions refer to the specific portions of a geometric figure that are highlighted for analysis, often to compute the area that is differentiated from the remainder of the figure. Techniques typically involve either adding up the areas of the shaded parts or subtracting the areas of non-shaded parts from a larger shape.

Transcript

00:01 When you look closely at this problem, you have y equals x, but then you also have the curve of y equals x squared minus 2.

00:11 Y equals x squared minus 2.

00:13 And it might not be clear where those two curves cross each other, but you can figure that out by setting them equal to each other.

00:21 So that would be x squared minus x.

00:24 Oops, that was supposed to say minus 2, wasn't it? i know i said that out loud, is equal to x.

00:34 So what you could do is you could subtract that x over and then factor this.

00:40 So negative 2 and positive 1 would give you factors that add to be negative 1.

00:45 So if you remember from your zero product property, that would be x equals 2 and x equals negative 1.

00:51 So the integral we need to do is from negative 1 to 2 based off the work we did there.

00:56 The upper function is y equals x minus the lower function which is x squared minus 2.

01:02 But when you distribute that minus to each piece, it becomes plus 2 dx.

01:09 So now we can do the integral of that.

01:11 So add 1 to the exponent, multiplied by the reciprocal of your new exponent.

01:18 And we're going from negative 1 to 2.

01:20 So plug in 2 for all those bounds, 2 squared is 4, half of which is 2, 2 cubed is 8, and then 2 times 2 will give you 4.

01:30 And then you have to subtract off...

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