Area of triangle help

[stacynk09]

I have NO IDEA how to work the following problem, our textbook contains no examples.

Use calculus to find the area of the triangle with the given vertices:
(0,5), (2, -2), (5,1)

Thanks!

 

[tkhunny]

Have you tried graphing the three lines defined by pairing up the three points two at a time? Do that first.

 

[stacynk09]

I plotted the three points to form a triangle.

 

[pka]

our textbook contains no examples.
Use calculus to find the area of the triangle with the given vertices:
(0,5), (2, -2), (5,1)

You do know how to write the equations of lines.
Graph the points.
Form two integrals: $\displaystyle \int\limits_0^2 {L_1 - L_2 } + \int\limits_2^5 {L_1 - L_3 }.$

 

[stacynk09]

Actually I'm not sure how to derive an equation from the points. Could you provide an example for me?

I believe that the equation for (0,5) is y=5-x

 

[galactus]

This is more of a Calc III method, but....

\(\displaystyle \L\\P_{1}=(0,5,0), \;\ P_{2}=(5, 1, 0), \;\ P_{3}=(2, -2, 0)\)

\(\displaystyle \L\\\vec{P_{1}P_{2}}=\langle5, -4, 0\rangle\)

\(\displaystyle \L\\\vec{P_{1}P_{3}}=\langle2, -7, 0\rangle\)

\(\displaystyle \L\\\vec{P_{1}P_{2}}\times\vec{P_{1}P_{3}}=\langle 0, 0, -27\rangle\)

\(\displaystyle \L\\\frac{1}{2}||\vec{P_{1}P_{2}}\times{\vec{P_{1}P_{3}}}||=\frac{27}{2}\)


To use integration, find the equations of your lines. If you can't do that, I am sorry to say, you're not ready for calculus. Just a check.
Once you have the line equations, integrate. You may have to break it into 2 regions.

 

[soroban]

Hello, stacynk09!

Use calculus to find the area of the triangle with the given vertices:
. . $\displaystyle (0,\,5),\;(2,\,-2),\;(5,\,1)$

You're in a Calculus course and you can't write the equation of a line?
. . Shame on you . . . go to your room!


Given two points $\displaystyle P(x_1,y_1),\;Q(x_2,y_2)$, can you find the slope of $\displaystyle PQ$ ?

. . Slope formula: \(\displaystyle \L\:m\:=\:\frac{y_2\,-\,y_1}{x_2\,-\,x_1}\)


Given a point $\displaystyle P(x_1,y_1)$ and a slope $\displaystyle m$, can you write the equation of the line?

. . Point-slope formula: $\displaystyle \:y\,-\,y_1\;=\;m(x\,-\,x_1)$

 

[stacynk09]

I believe I've figured out the equations. I would appreciate it if you could tell me if these are correct.

y=x-4
y=-3.5x+5
y=-0.8x+5

And thank you sooo much soroban.

 

[galactus]

Yep, that's correct. Good. Except, maybe write them with fractions instead of decimals. Looks better.

$\displaystyle y=x-4$
$\displaystyle \frac{-7x}{2}+5$
$\displaystyle \frac{-4x}{5}+5$