Find the area of the region (see figure) in two ways. (Figure cannot copy) a. By integrating wit

Find the area of the region (see figure) in two ways....

Key Concepts

Definite Integration

This concept involves computing the exact area under a curve or between curves by summing an infinite number of infinitesimally small slices. It forms the basis for many area calculations in calculus and is fundamentally linked to the Fundamental Theorem of Calculus, which connects differentiation and integration.

Integration with Respect to y

When integrating with respect to y, the region is sliced horizontally rather than vertically. This method can simplify area calculations, especially when the functions describing the region are easier to express as x in terms of y. It requires careful determination of the limits of integration along the y-axis and the setup of an appropriate integrand representing the horizontal 'thickness' of the slices.

Geometric Methods for Area Calculation

This approach involves breaking down a complex region into simpler geometric shapes such as triangles, rectangles, or circles where standard area formulas can be applied. It provides an alternative, often more intuitive, route to finding area without resorting to calculus. This method relies on recognizing familiar geometric patterns within the region of interest.

Transcript

00:01 So this problem wants us to find the area of the region by a integrating with respect to y and then b using geometry.

00:08 So if we're looking at part a where they want us to integrate with respect to y, it might be convenient to view our graph at a 90 degree angle.

00:19 That way we can think about it similarly that we would think about an area we were trying to find the region of if we were taking the integral with respect to x.

00:29 So we can start by determining our bounds.

00:32 So we can see that from 0 to 1 is our integration area space that we're dealing with.

00:43 And then you take the function that is above the other and you subtract those two.

00:48 So the higher one is y plus 1 and the lower one is 2y.

00:54 So we'll start with y plus 1 and we'll subtract that second function to y.

00:59 And this is all with respect to y.

01:03 So then performing some simple algebra, we can simplify this down to 1 minus y, d ,y, and then we can go ahead and integrate.

01:12 Using some integration rules, we get 1 integrates to y, and negative y integrates to negative y squared over 2, where this is evaluated from 0 to 1.

01:25 So we go ahead and plug in the one first and then subtract this from the value we get by plugging in zero.

01:38 This gives us one minus one half minus zero and this simplifies down to just a half.

01:50 So then for part b, they want us to integrate this using some algebra.

01:57 So if we look at this large triangle, this like invisible large triangle we could draw, that's kind of a bad representation.

02:06 Let me read about that.

02:10 It's going to be difficult for me to delete it, but i shall try.

02:16 Let me try that one more time.

02:17 I'm going to use a different color, though.

02:19 Let's do purple.

02:21 So we have this triangle here, this purple triangle...