Find the areas of the regions enclosed by the lines and curves. y=|x^2-4| and y=(x^2 / 2)+

Find the areas of the regions enclosed by the lines and...

Key Concepts

Area Between Curves

This concept involves finding the region enclosed by two or more curves by integrating the difference between the functions (the upper function minus the lower function) over the interval determined by their points of intersection. It is a fundamental application of definite integrals in calculus.

Intersection Points

Determining the intersection points of the curves is essential because they define the limits of integration. Solving for these points ensures that the area calculation accurately represents the region enclosed by the curves.

Absolute Value Functions

When a function includes an absolute value, it requires careful handling since the absolute expression may change the behavior of the function over different intervals. Understanding and rewriting the absolute value function as a piecewise function is necessary for proper analysis and integration.

Piecewise Function Analysis

Converting functions with absolute values or other conditions into piecewise forms allows for correct integration over intervals where the function's expression changes. This analysis is crucial for ensuring that each segment of the function is treated with its proper formula in the area calculation.

Transcript

00:02 So we want to find the areas of the regions enclosed by these two curves.

00:06 First, let's just try to maybe draw a picture.

00:16 So what does this one look like? the absolute value of x squared minus 4.

00:21 Well, it basically looks like just the graph of x squared minus 4, so a parabola that's been moved downward by four units.

00:33 But wherever that graph is below the x -axis, we're going to flip it up, so it's above the x -axis.

00:39 So that graph would actually look something like this.

00:44 So when x is equal to 0, it would be at y equals 4.

00:54 It would go out here to negative 2 and 2.

00:58 That's where it's 0, and then otherwise it would be going up like this.

01:06 So imagine just the parabola x squared minus 4, where this bottom bit is reflected above the x -axis.

01:16 Okay, and so we also have this.

01:19 Parabola and we want to find where these two are intersecting.

01:27 So let's just find where is x squared minus 4 equal to x squared over 2 plus 4.

01:43 So if we just move the x squared to the left, we have x squared over 2 on the left is equal to 8, so x squared is equal to 16 means that x is plus or minus.

02:02 So here's 2 somewhere out here should be 4.

02:09 Maybe i didn't draw this very accurately at first, but it should be going up like this maybe.

02:24 Okay, and so then we have this point here, and let's see, when x is equal to 4 here, let's check that this actually makes sense.

02:40 So if i plug in x is equal to 4, here i get 12, okay, and here i get 16 over 2, that's 8 plus 4 is 12.

02:50 Okay, great.

02:51 So somewhere they intersect up here beyond where i've, above here, somewhere where y is equal to 12.

03:03 So to find the area, we're just going to, again, sort of by the symmetry, maybe we can just integrate from 0 to 4 and double it to get what's on the left hand side.

03:16 So the area i want to say is twice the integral from 0 to 4 of x squared over 2 plus 4 minus the absolute value of x squared over 4 dx.

03:43 Okay, well now to integrate this part of x squared minus 4 of the absolute value, we sort of need to break it up...

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