# Units of the integral (x=seconds, y=metres)

• bingoboy
In summary, the area under the graph of velocity vs time (in meters per second) would be expressed in meters per second.
bingoboy

## Homework Statement

I'm plotting a man jumping over a car for maths and I'm trying to find the critical velocity at which the car has to travel for the jumper to make it safeley over the car the man's jump is approximated by the parabola y= -0.0176x^2 + 0.3595x - 0.2794 and restricted by the height of the car y=1.225

So basically I have units of metres on the y-axis and seconds on the x-axis and i want to find the area under the curve of y= -0.0176x^2 + 0.3595x - 0.2794 and restricted by the line y=1.225. However i know how to do all the Integral stuff the only trouble I'm having is with what the units will be for my answer

## Homework Equations

units of y-axis = metres
Units of x-axis = seconds

## The Attempt at a Solution

My best guess is the units will be m/s which is what will be of most use to me, however if it's not is there a way i can manipulate the graph to give me an answer in metre's per second i.e instead of t make it t^2 or somthing like that

In a graph of meters vs seconds, the slope at any point (the derivative, in this case velocity) will be expressed as simply meters/second [m/s]

Yes but that will give me the instantaneous velocity of the jumper won't it? I think I'm trying to be too tricky with this problem

Well, velocity is always m/s, whether it's critical, instantaneous, or the magnitude (ie. speed).

But I think I understand what you are getting at. Suppose you had a graph with m on the y-axis and s on the x axis. You want to know what the unit would be for the area under the graph?

I guess you can just look at the graph from a geometric standpoint. If the integral is essentially adding up the area under the graph, let's consider a graph like this:

Only let's say that the y-axis is in some unit m and the x-axis is in some unit s. Geometrically, when you add up the area under that graph using the formula for area of a rectangle (L*W), what will the units be?

Now recall, that during integration, you are simply separating the space under a curve into little segments dx (well, ds in this case) and multiplying that by the height (y, or m in this case). Can you see what the units would be now?

## What are the units of the integral?

The units of the integral depend on the units of the variables being integrated. For example, if x is measured in seconds and y is measured in metres, the units of the integral would be seconds times metres, or seconds-metres.

## Can the units of the integral be changed?

Yes, the units of the integral can be changed by using appropriate unit conversions. This can be done by multiplying or dividing the integral by a conversion factor.

## Why is it important to pay attention to units when using integrals?

Units are important in integrals because they represent the physical quantities being integrated. Using incorrect units can lead to incorrect results and misinterpretation of data.

## What happens to the units when taking the definite integral?

The units of the definite integral are the same as the units of the original function being integrated. However, the units may change if the limits of integration have different units than the variable being integrated.

## Can the units of the integral be used to determine the physical meaning of the integral?

Yes, the units of the integral can provide insight into the physical meaning of the integral. For example, if the units of the integral are metres, the integral represents the total distance traveled over a given time period.

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