Use the three-step procedure (slice, approximate, integrate) to set up and evaluate an integral

Use the three-step procedure (slice, approximate, integrate)...

Key Concepts

Region Slicing

This concept involves visualizing or partitioning the area of interest into thin, manageable slices or elements (typically rectangles). It requires deciding whether to slice the region vertically, horizontally, or along another variable, and identifying the dimensions (such as height and width) of each slice based on the geometry of the region.

Riemann Sum Approximation

This concept approximates the total area by summing the areas of individual slices. Each slice's area is approximated (for instance, by multiplying its height by a small width), and these approximations form a Riemann sum that represents the area under a curve or of a region.

Definite Integral Formation

This concept rests on taking the limit of the Riemann sum as the slice width tends to zero, which transforms the summation into a definite integral. The integral, with appropriate bounds corresponding to the limits of the region, exactly computes the area by accumulating the contributions of infinitely many infinitesimally thin slices.

Transcript

00:02 We have a region shown by two graphs and x equals negative 2 and x equals positive 2.

00:11 So in that region, we are going to find the area and we're going to represent one slice.

00:17 We're going to approximate the area and then we're actually going to go ahead and integrate.

00:22 So for our little slice, you know, it is a rectangle and it is, has a thickness of dx.

00:30 So remember that dx is going to zero.

00:33 So when we have all those rectangles all the way from negative 2 to 2, it will actually fill in the area between the two curves.

00:41 So notice the reason why we draw this slice is to see what is the top function and what is the bottom function.

00:48 So you can see that your quadratics on top and then your linears on the bottom.

00:53 So to approximate that area, what we're going to do is we're going to just make some geometric shapes.

00:59 And decided to make all triangles for this situation because a triangle is kind of closest.

01:05 I can see that our triangle is a little bit of an overestimate, but we have two triangles that are the same on top, and then we have the triangles that are the big triangle at the bottom.

01:19 And so we're looking for when we finish our integration for our answer to be close to 16.

01:25 Okay, i'm going to go ahead and get rid of those triangles, though.

01:29 So we can actually see kind of what we're working with here.

01:34 So we'll just erase those as good as we can without erasing our slice here.

01:41 Okay.

01:43 So now we're going to be writing our integral.

01:46 So we're going to go from negative 2 to 2.

01:49 And then we said that the quadratic is the function that's on top...