# 7f. Area Under a Curve - Definite Integration

## Integration Mini Video Lecture

Historically, areas between curves were a hot problem and inpsired the development of integral calculus. This video gives an overview on how to use integration to find an area under a curve.

## Video transcript

[Music...]

### Related lessons on IntMath

This 'Area under a curve' mini video lecture explains the processes in these two pages:

3. Area Under a Curve (in this chapter)

... and ...

Area Under a Curve by Integration (in the Applications of Integration chapter)

Area Between Two Curves by Integration (also in the Applications of Integration chapter)

Area under curve definite integral... a mini lecture.

One of the first applications of integration was to find the area under a curve.

Though there were approximate ways of finding this, nobody had come up with an accurate way of finding an answer [until Newton and Leibniz developed integral calculus].

So let's have a look at this example.

Find the area under the curve *y* equals 2*x*^{3} + 5 between *x* equals 1 and *x* equals 2.

Let's see what we are talking about here.

OK, here' the curve *y* equals 2*x*^{3} + 5 and here is *x* equals 1 and *x* equals 2.

So what we are doing is to find this area in here.

Now the formula for area is integral *a* to *b* of function *x* [*f*(*x*)] *dx*.

Well in this particular example, our function is *y* equals 2*x*^{3} + 5, so I copy this...

And.so we can say that the area equals:

The integral from 1 to 2 ... of 2*x*^{3} + 5 *dx*.

Thats the... 2*x*^{3} + 5 in brackets. Don't forget the brackets, that's always important.

Now, how do we do this problem?

Well what I must do is integrate this thing, and substitute 2 in and then minus substitute 1 in. That's how we do a definite integral.

So, first step, put square brackets [ ]. And then integrate... [*x*^{4}] plus 5 times *x*, lower value is 1, upper value is 2.

It's going to be 2 and then its going to be *x* to the power of 4 because we are addding one to the index and then dividing by the new number.

Plus 5 times *x* because the integral of 5 is 5*x*.

Lower value is 1 and upper value is 2.

Then this equals... The 2 and the 4 cancel..so

Let's cheat a little. Copy and paste this. And cancel out the 2 and 4 [to become 2 in the denominator].

So great! I have done the integration and now what I need to do is to do the substitution steps. So it's going to look like this.

I am going to copy this bit, and paste it, minus, paste it.

And substituing in the 2 [in the power 4 expression] ,and then [plus] 5 times 2 ... minus substituting in the 1 [in the power 4 expression] and [Minus] 5 times 1.

So, just to tell you what I did again, I integrate this expression 2*x*^{3} + 5 and I get *x* to the power 4 on 2 plus 5*x*. `(int (2x^3+5) dx = x^4/2 + 5x)`.

Then for definite integral we substitute 2 into this, and then 1 into this to get this and then using Scientific Notebook to give me the answer.

I get 25 over 2, or 12.5. (Here's the calculation:)

`int_1^2 (2x^3+5) dx ` `= [x^4/2 + 5x]_1^2 ` `= 25/2 ` `{:= 12.5`.

And of course we don't know what the units are. But we need to put something like "units squared" to indicate we know it's an area.

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