# Show the area under a curve

## Homework Statement

Show the area under the curve of v(t) is equal to the displacement from t1 to t2

x/t = v

## The Attempt at a Solution

Integrate V(t) = vt dt
(v/2)*t^2]t1 to t2
(v/2)*t1^2 - (v/2)*t2^2

Not sure if that is good enough or how toactually show it. To find the area you take the integration and v(t) is just the derivative of x(t) but im not how to show it exactly.

I would do it like this:
##\int^{t1}_{t2} v(t)dt##
You basically have it.

Thank you how do we write like that in this forum?

Mark44
Mentor

## Homework Statement

Show the area under the curve of v(t) is equal to the displacement from t1 to t2

## Homework Equations

x/t = v
This equation isn't relevant if the velocity isn't constant.
brycenrg said:

## The Attempt at a Solution

Integrate V(t) = vt dt
(v/2)*t^2]t1 to t2
(v/2)*t1^2 - (v/2)*t2^2

Not sure if that is good enough or how toactually show it. To find the area you take the integration and v(t) is just the derivative of x(t) but im not how to show it exactly.
Since v(t) = ##\frac{dx}{dt}##, your integral is ##\int_{t_1}^{t_2}v(t) dt = \int_{t_1}^{t_2} \frac{dx}{dt} dt = \int_{t_1}^{t_2} dx##. If you carry that out, what do you get?

thank you guys. You get t2-t1

haruspex
Homework Helper
Gold Member
2020 Award
thank you guys. You get t2-t1
No. In the final integral in Mark's post, the limit variable and integration variable are different: ##\int_{t=t_1}^{t_2}dx##.
What is x when t=t1?

Well isn't x = t1 when t is t1
I thought it was x]t2 upper t1 lower
So it's t2 - t1

haruspex
Homework Helper
Gold Member
2020 Award
Well isn't x = t1 when t is t1
I thought it was x]t2 upper t1 lower
So it's t2 - t1
No. x is a position. What is the position at time t1? (so create one!)

So I could say t1 = 1 and t2 = 2
So then it would be 1 in that case.
So the area would be 1 lol I dono

haruspex
Homework Helper
Gold Member
2020 Award
So I could say t1 = 1 and t2 = 2
So then it would be 1 in that case.
So the area would be 1 lol I dono
No, you can't just plug in arbitrary numbers.
The question asks you to show that the area equals "the displacement from t1 to t2". If the displacement x is a function of t, x(t), how would you write the displacement at time t?

X(t2) - x(t1) is that what they want?

haruspex
• 