Use calculus to find the area of the triangle with the given vertices. (2,0),(0,2),(-1,1)

Name: Problem 30, Chapter 5: Calculus
Uploaded: 2019-12-24T19:14:12-08:00
Duration: 5 min 2 s
Channel: Kenneth Kobos
Description: Use calculus to find the area of the triangle with the given vertices. (2,0),(0,2),(-1,1)

Use calculus to find the area of the triangle with the given...

Key Concepts

Double Integration

Double integration extends the idea of definite integrals to functions of two variables. It is used to compute areas (or volumes) by integrating over a region in the plane. This approach requires setting up appropriate limits for the variables according to the boundaries of the region.

Definite Integration

Definite integration is the process of evaluating the area under a curve between two specific limits. In the context of finding areas of regions in the plane, it involves summing infinitely many infinitesimal contributions, which is a fundamental technique in calculus applied to determine the size of a space bounded by functions.

Green's Theorem

Green's Theorem provides a powerful connection between a line integral around a closed curve and a double integral over the region enclosed by that curve. It offers an alternative method to compute the area of a region, like a triangle, by converting the area calculation into an integral along the boundary of the region.

Transcript

00:01 So we have these three points which are going to form a triangle and we want to determine the area of the enclosed triangle.

00:07 So let's plot the points to 0, 02 minus 1, 1.

00:13 Now we connect them to form a triangle.

00:21 And we want to determine the area of this enclosed region here.

00:25 To do that, we're going to need the equations of the lines.

00:29 So this one over here is y equals, the slope is 1.

00:33 So y equals x and the y intercept is 2.

00:37 This bottom one here, y is equal to.

00:40 The slope here is negative 1, so negative x, and again the y intercept is 2.

00:45 And this last one is y is equal to.

00:49 The slope here is negative 1 over 3x plus some b value.

00:55 So let's figure out the b value.

00:57 We know y equals negative 1 3x plus b.

01:01 We can plug in any point on this line so for example we have this point to zero so we can just plug in 0 equals negative 1 over 3 times 2 plus b and then what we get is 2 3 equals b okay so negative 1 3 plus 2 over 3 great now what we want to do is integrate to figure out the area area is equal to integral so first we're going from x equals negative 1 to x to x equals 0 it's going to be top function minus bottom function the top function is x plus 2 x plus 2 minus the bottom function which is negative 1 3 3 3 3 1 3 3 x plus 2 over 3 d x plus now we're integrating from 0 to x equals 2 2 it's going to be the top function minus the bottom function the top function the top function is negative x plus 2 over here that's negative x plus 2 negative x plus 2 negative x plus 2 and then we're subtracting the bottom function which again is the line negative 1 3 3x okay once we calculate all of this we'll have the area so let's just simplify things a little bit first integral negative 1 to 0 here we have an okay let's use a different color here we have x plus one third x, so that works out to four thirds x.

02:57 Then we have a plus two, and then we have a minus two thirds.

03:02 So that works out to plus four over three, dx, plus integral from zero to two.

03:12 Let's simplify this a little bit.

03:13 We have a negative x plus a one third x that works out to negative two thirds x...

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