Find the area of the triangle with vertices at A(0,0),B(1,7),and C(5,4).

The area of the triangle is the area between `bar(AB)` and `bar(AC)` from 0 to 1 plus the area between `bar(BC)` and `bar(AC)` from 1 to 5.

The equation of the line going through A and B is y=7x.

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Find the area of the triangle with vertices at A(0,0),B(1,7),and C(5,4).

The area of the triangle is the area between `bar(AB)` and `bar(AC)` from 0 to 1 plus the area between `bar(BC)` and `bar(AC)` from 1 to 5.

The equation of the line going through A and B is y=7x.

The equation of the line going through B and C is `y=-3/4x+31/4`

The equation of the line through A and C is `y=4/5x` .

Thus the area of the triangle is given by:

`A=int_0^1(7x-4/5x)dx+int_1^5(-3/4x+31/4-4/5x)dx`

`=int_0^1 31/5xdx+int_1^5(-31/20x+31/4)dx`

`=31/10x^2|_0^1-31/40x^2|_1^5+31/4x|_1^5`

`=31/10-(155/8-31/40)+(155/4-31/4)`

`=31/2`

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**The area of the triangle is 15.5 square units.**

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Check: Another way to find the area of a triangle given its coordinates --

`A=+-1/2|[0,0,1],[1,7,1],[5,4,1]|=31/2` as above.

The graph: