Use calculus to find the area of the triangle with the given vertices. (0,0),(3,1),(1,2)

Name: Problem 29, Chapter 5: Calculus
Uploaded: 2022-10-07T17:47:28-08:00
Duration: 4 min
Channel: Adalynn Griesser
Description: Use calculus to find the area of the triangle with the given vertices. (0,0),(3,1),(1,2)

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

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus used to compute the accumulation of quantities, such as areas under curves. When finding the area of a region, the definite integral sums up the infinitely small slices (or strips) of area across an interval, providing a precise measurement of the area between the boundaries.

Setting Up the Region of Integration

Setting up the region of integration involves determining the limits over which the integration will occur. This process requires identifying the outer boundaries of the shape, such as curves or lines, and understanding how these boundaries interact to enclose a region, so that the integral correctly captures the entire area of interest.

Equations of Lines from Two Points

Finding the equation of a line given two points is essential when the boundaries of a region are defined by straight lines. By using the coordinates of the points, one can derive the slope-intercept form or other forms of the line equation, which then serve as the limits or integrands in setting up the area integral.

Piecewise Integration

Piecewise integration is used when the region to be integrated is divided by different functional boundaries across its domain. In such cases, a single integral may not be sufficient to cover the entire region, so the area is calculated by summing the separate integrals over the different intervals where the boundary function changes.

Transcript

00:02 In this question, we're asked to find the area of this triangle using calculus.

00:08 Now, i've just named these sides of triangle a, b, and c, because we're going to have to write equations for these sides of the triangles.

00:16 So side a is given, well, first let's find the slope of this line, which would be from 2, 1, minus 0 ,0.

00:32 So the slope of that line is 2, the y intercept is 0, so that equation is y equals to x for side b from 1 comma 2 to 3 comma 1.

00:46 If we take the slope of that, that is negative 1 half.

00:54 So y equals negative 1 half plus x plus it has some other y intercept.

01:03 We know that one or three comma one is a point on this triangle so we plug that in and b equals five halves so y equals one half x plus five halves and for side c that's from zero zero to one comma two um or no from three zero to three comma one that is one minus zero over three minus zero that's the slope of one -third.

01:36 The y intercept is 0, so y equals 1 third x.

01:42 Now to find the area of this triangle, we're going to have to take an integral, but we have pointy bits here at the points of the triangle, so we have to split it at some point.

01:52 Because our equations are written in terms of x, we're going to integrate with respect to x, and i'm going to split the triangle right in half -ish right here at x equals 1.

02:04 So the integral is the integral from x equals 0 to 1 of the top equation minus the bottom equation...

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