Sketch the region enclosed by the given curves. Decide...

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $x$ or $y$. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.
$y=x^2-2 x, \quad y=x+4$

Key Concepts

Curve Sketching

This involves drawing the curves to understand the geometric region they enclose. Sketching aids in visualizing the area and the relative positions of the curves, which is crucial for setting up the proper integration limits and identifying the top and bottom (or right and left) functions.

Finding Intersection Points

The points where the curves intersect determine the boundaries of the region. Solving the equations of the curves simultaneously gives these intersection points, which then serve as the limits of integration when setting up the definite integral.

Choosing the Variable of Integration

Deciding whether to integrate with respect to x or y depends on the orientation and shape of the region. One chooses the variable that simplifies the integral's bounds and the function expressions that represent the top/bottom or left/right sides of the region.

Understanding Approximating Rectangles (Differential Elements)

Drawing a typical approximating rectangle represents a differential element of area. Labeling its width and height helps conceptualize how an integral sums these small pieces to approximate the total area. This is the foundation of the Riemann sum approach to integration.

Setting Up and Evaluating the Definite Integral

Once the region, the intersection points, and the appropriate variable of integration are determined, one writes a definite integral that represents the area. This involves taking the difference between the upper and lower (or right and left) functions over the interval and then evaluating the integral to find the exact area.

Transcript

00:01 Okay, we want to find the area between these two curves.

00:03 The first one is a parabola.

00:05 So let's complete the square so that we can see where to put it.

00:10 Half of negative two is negative one.

00:12 Negative one squared is plus one.

00:14 And you can't go around.

00:15 Just add one to stuff.

00:16 So you've got to subtract one also.

00:19 So that's x minus one squared plus one.

00:25 All right.

00:26 So right one, up one.

00:30 Then one, one.

00:33 There it is.

00:34 Oops.

00:36 All right.

00:37 Y equals x plus 4 it's a line slope is positive one so up one over one okay so it looks like this all right so you can see that this would be much easier to integrate with respect to why because no matter which slice i draw they will all have the green line on the top and the red curve on the bottom this one this one this one this one it doesn't matter they all have the same top and bottom, so that's the way to go.

01:15 So the area is going to be x on the left to x on the right, y top, minus y bottom, dx.

01:27 All right, so i need to know what these two points are.

01:30 My drawing's not that good, so i'm going to have to work it out...

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