Use calculus to find the area A of the triangle with the given vertices. (0, 0), (5, 6), (1, 8) .

Question

Use calculus to find the area A of the triangle with the given vertices. (0, 0), (5, 6), (1, 8)

Answer 1

Always start this kind of problem by drawing a picture....

The triangle is widest along the y-axis, so I think integrating in that direction will easiest.

The are between 2 functions is the difference between them, evaluated over the appropriate interval. In this case, we will integrate from y = 0 to y = 8.

Now, we just need x as a function of y....

At the bottom is a line from (0,0) to (1,8). The equation of this line is x = y/8

At the top, we have 2 zones:

From y = 0 to 6, the line is x = 5y/6

From y = 6 to 8, the line is x = 5 - 2(y - 6)

So now we do the integration in 2 stages:

A₁ = ∫ (5y/6 - y/8), from 0 to 6

= 5y² /12 - y² /16

= 15 - 36/16 = 12.75

A₂ = ∫ (5 - 2(y - 6) - y/8)

= ∫ -17y/8 + 17, from 6 to 8

= -17*y² /16 + 17y

= -17*64/16 + 136 + 17*36/16 - 102

= 34 - 17*28/16 = 4.25

Total area = 17