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

(0, 0), (3, 1), (1, 8)

A = ????

the line through (3,1) and (1,8) is 7x+2y = 23

the line through (0,0) and (1,8) is y = 8x

the line through (0,0) and (3,1) is y = 1/3 x

So, the area is

∫[0,1] (23-7x)/2-x/3 dx + ∫[1,3] 8x-x/3 dx

no sweat, eh?

See the lines at

http://www.wolframalpha.com/input/?i=plot+7x%2B2y%3D23%2Cy%3D8x%2Cy%3D1%2F3+x+for+x+%3D+-2..4

Ah, the area of a triangle! Is there an easier way to find it? Maybe we can use pick-up sticks or count the number of cheerios we can fit inside. Either way, let's crunch some numbers!

To calculate the area of a triangle using calculus, we'll use the formula:

A = 1/2 * |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))|

Plug in the values for the vertices:

A = 1/2 * |(0(1 - 8) + 3(8 - 0) + 1(0 - 1))|

Now let's simplify:

A = 1/2 * |(0 - 0 + 1)|

A = 1/2 * 1

A = 1/2

So, the area of the triangle is 1/2 of whatever unit you're using to measure. Keep in mind that the answer is always positive, regardless of orientation.

To find the area of a triangle using calculus, we can use the formula:

A = 1/2 * |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|

Given the vertices (0, 0), (3, 1), and (1, 8), we can substitute the coordinates into the formula. Let's calculate it step-by-step:

x₁ = 0, y₁ = 0
x₂ = 3, y₂ = 1
x₃ = 1, y₃ = 8

A = 1/2 * |0(1 - 8) + 3(8 - 0) + 1(0 - 1)|

Simplifying the expression inside the absolute value:

A = 1/2 * |-7 + 24 + 0|
A = 1/2 * |17|

Since the absolute value of 17 is 17, we have:

A = 1/2 * 17
A = 17/2
A = 8.5

Therefore, the area of the triangle with vertices (0, 0), (3, 1), and (1, 8) is 8.5 square units.

To find the area of a triangle using calculus, we can use the formula:

A = 1/2 * |(x1 * (y2 - y3)) + (x2 * (y3 - y1)) + (x3 * (y1 - y2))|

First, let's assign the coordinates of the vertices to variables:

(x1, y1) = (0, 0)
(x2, y2) = (3, 1)
(x3, y3) = (1, 8)

Substituting these values into the formula, we have:

A = 1/2 * |(0 * (1 - 8)) + (3 * (8 - 0)) + (1 * (0 - 1))|

Simplifying the expression inside the absolute value, we get:

A = 1/2 * |(0 * (-7)) + (3 * 8) + (1 * (-1))|
= 1/2 * |0 + 24 - 1|
= 1/2 * |23|

Since the expression inside the absolute value is positive, we can remove the absolute value sign:

A = 1/2 * 23
A = 11.5

Therefore, the area of the triangle with the given vertices is 11.5 square units.