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

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

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Key Concepts

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.

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.

Transcript

00:01 Given the region shown, with the equation shown, we are going to show a slice for integration of the area.

00:09 We're going to approximate that area, and then we're actually going to integrate.

00:15 So notice our slice is a very small rectangle that has a thickness of dx.

00:24 And remember, when you have an infinite amount of these rectangles, that's what your integration is.

00:32 So you're finding the area between the curve shown and the x -axis, which is also considered y equals zero.

00:39 So let's go ahead and just kind of approximate that area.

00:43 You can kind of just make shapes to give yourself, you know, an estimate so that you know that your answer is correct.

00:50 So when i drew the rectangle, there's as much area kind of over it as under it.

00:56 So i feel like that's pretty good.

00:58 And then the triangle's pretty close to the curve.

01:00 You can see that the triangle is a little bit bigger, but it's a good estimate.

01:05 So our rectangle and then our triangle.

01:09 And so we're looking for our area to be, you know, close to that 9 .5.

01:15 So we'll see how close we get.

01:17 Okay, so to actually integrate, we need to consider we have a start value and an end value.

01:23 And let's see, let's go ahead and get rid of our old shapes here...

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