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

step 1 We're given a region here. The region is between y equals zero and the function given the quadra

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

00:02 We're given a region here.

00:03 The region is between y equals zero and the function given the quadratic.

00:10 So we are going to represent a slice.

00:13 We're going to approximate that area and we are going to integrate.

00:18 So considering that this is the region, let's go ahead and do our approximation first.

00:24 Now, i'm going to be doing triangles, well, one triangle.

00:29 If you can kind of see that if you straighten down, those lines.

00:33 So our base is four because we're going from negative three to one and then you can see that the height is also four.

00:41 So we're thinking that a good approximation for the area would be eight.

00:46 Okay, so one of the slices would be, you know, represented here.

00:52 And really the reason to do a slice is to understand who the top function is and who the bottom function is.

01:00 So what we have here is we have a function y equals zero.

01:04 That's the actually touches the top of our slice where our quadratic that's given touches the bottom.

01:11 So we are going to have to do zero minus the function given.

01:16 So what this really means is we're going to distribute the negative and everybody will end up changing sign.

01:22 All the terms will change sign.

01:24 So we'll have a negative x squared, a negative 2x, and then a positive 3...

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