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 A region is shown and the equations used to create that region are also shown.

00:08 What we're going to do is we're going to find the slice for integration.

00:13 We are going to approximate the area and we are going to integrate.

00:17 So we're not necessarily going to go in the order of what it's asked us to.

00:20 We're going to do the approximation first.

00:23 So what i decided to do is i decided to draw an actual triangle.

00:29 Now notice i left out some area, but i feel like it's pretty close to the area that i included.

00:36 So with your shapes, again, this finding of an area is more about getting a good estimate so that you know whether your answer is correct or not.

00:47 And so you can see that the base and the height of my triangle are three.

00:52 So half a nine would be about 4 .5.

00:57 Okay, let's go ahead and get rid of that.

00:59 So we can see our actual region.

01:02 Okay, so now we need to draw our slices.

01:06 So, and actually before i do that, i'm going to go ahead and find my intersections.

01:15 So when we're actually going to go to integrate, we are going to have to integrate from some value to another value.

01:23 Now notice, i have one equation that says y equals and one equation that says x equals.

01:29 Well, the equation that's x equals is a quadratic.

01:33 So we don't call this a function because notice if we did a vertical line, we would actually get our two values of y for one value of x.

01:43 So we're going to call this a curve, or we still can call it a parabola.

01:49 But what we notice here is that it's going to be a lot easier to stay in terms of y.

01:57 So this x equals is the best form.

02:00 If we wanted to solve for y equals in terms of x, we would have the whole plus or minus.

02:06 So all of a sudden we'd have two of functions and we have to be dealing with three overall.

02:10 So i'm going to go ahead and set this equation equal to the other one.

02:15 Well, in order to do that, that also has to be x equals an equation in terms of y.

02:22 So that's going to be a y, and then we just add over the one...

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