Use a graph to find approximate x-coordinates of the points of intersection of the given curves.

Use a graph to find approximate x-coordinates of the points...

Key Concepts

Points of Intersection

Points of intersection are the solutions to the system of equations formed by two or more functions. These are the points where the graphs cross each other and are key to understanding where the functions have the same output value for a common input value, serving as boundaries for regions of interest.

Numerical Approximation

Numerical approximation methods are techniques used to estimate values when exact solutions are difficult to obtain analytically. In the context of graphing and integration, these methods help estimate points of intersection and areas under curves by relying on graphical analysis or numerical integration techniques.

Definite Integration

Definite integration is the process of evaluating the integral of a function between two specific limits. It provides a numerical value representing the accumulated area under a curve or between curves, making it an essential tool in calculating the exact or approximate area of regions.

Graph of Functions

Graphing functions involves plotting the set of all points (x, y) that satisfy the given equations on a coordinate plane. This process is critical for visualizing the behavior of functions, their relative positions, and to identify notable features such as intersections, turning points, and symmetry.

Area Between Curves

The area between curves is a concept in integral calculus that refers to the region enclosed by multiple function graphs. It is calculated by integrating the difference between the top function and the bottom function over the interval defined by their points of intersection.

Transcript

00:01 We want to find the area of the region enclosed by these two functions.

00:04 The first one in blue, y equals 3x squared minus 2x, and the second in green.

00:09 Y equals x to the 3 minus 3x plus 4.

00:13 So if we do a quick graph using a graph in calculator, at first glance it looks like there's one region over here.

00:22 Because at first glance it looks like there's two intersection points.

00:25 But if we actually zoom out, we find that there's a third intersection point, which happens at a higher x value.

00:34 So we actually have two regions, the first one being down here, which we saw on the first slide, and the second region is this tiny sliver that goes over here.

00:52 So the total area is actually going to be the sum of two areas, which will solve independently.

00:57 So i'll write this as area is equal to area 1 plus area, so we're going to figure out area 1 and area 2 separately.

01:11 So area 1 i drew it in green, and area, i mean in red, and area 2, i drew in black.

01:19 So let's solve these two areas separately.

01:23 Area 1 is equal to, we're going to do an integral.

01:28 So let's look at our first picture, since it's a bit more clear.

01:32 Our first x value is going to be negative 1 .115, and our upper value is going to be.

01:39 To be 1 .254.

01:42 So let's write those in.

01:45 Negative 1 .115 to 1 .254.

01:54 And it's going to be the upper function minus the lower function.

01:57 Here the upper function is the green one and the lower function is the blue one.

02:02 So we're going to subtract those.

02:08 The green function, x cubed minus 3x plus 4, minus the blue function, 3x squared minus 2x dx.

02:27 Okay, of course we can simplify this a little bit.

02:34 We have an x cubed, we have minus 3x squared, we have a minus x and then we have a plus 4 dx.

02:45 So we can carry out this integral by doing the antiderivative, 1 fourth x to the 4 minus x cubed...

Please give Ace some feedback