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

Key Concepts

Intersaction of Curves

This concept involves solving equations to determine the points at which curves intersect. Finding these intersection points is crucial to defining the limits for integration, as they represent the boundaries of the region whose area is being calculated.

Area Between Curves

The area between two curves is found by integrating the difference between the upper and lower functions over a specified interval. This method uses the concept of adding the 'slices' of the region to measure the total area enclosed.

Definite Integration

Definite integration is a fundamental calculus technique used to sum an infinite number of infinitesimally small quantities over a specific interval. It is employed here to compute the area under a curve or between curves by evaluating an integral between the limits determined by the intersection points.

Graphical Analysis of Functions

Graphical analysis involves sketching the functions to visually interpret the behavior and relative positions of curves. This helps in understanding the region of interest and in setting up the correct integral by identifying which function is above or below the other within the region.

Transcript

00:01 This problem isn't too bad because you can actually just do the y equals x squared on its own.

00:09 And if you think about y equals, i'm doing this in red because it's not going to match up very well, x to the fourth minus three x squared, that if you were to factor out in x squared, and then you'd be left with x squared minus three, that you can see that x equals zero is going to bounce off.

00:32 The x -axis it would be better if i said it's tangent at x equals zero and then the graph will go up and i did not draw my i did not draw my parabola very well because this other one is going to shoot up pretty quickly in here same thing with this one so there we go so the question that becomes is where are these points of intersection which you can find by actually setting the equations equal to each other and the fastest way to do this would be to subtract x squared to the right side.

01:10 So you have minus 4x squared, and then again, factor out in x squared, x squared minus 4.

01:18 But now you can see that x equals 0 is a point of intersection.

01:22 But because it's tangent there, we can actually ignore that when i set up my integral, and then positive in negative 2.

01:28 That's how we would get answers of 0 on this problem.

01:31 So i'm just going to set up my integral from negative 2 to 2 of my upper function being x squared, and then i'm i have to subtract off the lower function of x to the fourth minus 3x squared dx, and just make sure that you distribute that minus into the problem.

01:49 Negative 2 to 2 of negative x to the 4th.

01:53 That becomes a plus 4x squared, dx.

01:58 So to evaluate this integral, add 1 to the exponent, multiplied by the reciprocal...

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