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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

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

( 0 , 0 ) ( 3 , 1 ) ,   ( 1 , 2 )

To determine

To find:

The area of the triangle with the given vertices, using calculus.

Explanation

1) Concept:

Formula:

The area A of the region bounded by the curves y=f(x), y=g(x) and the lines x=a and x=b is

A= abfx-gxdx

fx-gx=fx-gx when fxg(x)gx-fx when gxf(x)

Equation of line passing through two points (a,b) and (c,d) is given by  y-bx-a=d-bc-a

2) Given:

The vertices 0, 0,  3, 1,  (1, 2)

3) Calculation:

First, find the equations of lines passing through the given vertices.

Equation of the line passing through (0, 0) and (3, 1) is

y=13 x

Equation of the line passing through (0, 0) and (1,2) is

y=2x

And the equation of the line passing through (3, 1) and 1, 2 is

y= -12 x+52

Now, use these equations to find the area enclosed by the given vertices.

The area between x=0 and x=1 is bound by y=2x and y=13 x

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