Don't quite understand a concept of integration - conflicting concepts - need help! (1 Viewer)

[blackops23]

Hi, I just started integration a few days ago, and I'm up to the part of the Definite Integral and finding an area below the x-axis and all that stuff. Problem is, the Cambridge definition of definite integral with an integrand with negative values, is COMPLETELY different from the booklet my school teachers gave me (which includes explanations from J&C).

Basically here's the problem, I'm going to use an example to illustrate it.

f(x) = x^3 - 4x
Now I must integrate this in the bounds of x=2 and x=-2
As you can see for the graph, for -2(<)x(<)0, f(x) is positive, and for 0(<)x(<)2, f(x) is negative.

By the cambridge method, integration was done as normal, and the answer was 0.
By my booklet's method, the integration was split into two sections, one for the positive area and one for the negative area, except the absolute value of the negative area was taken.

So by cambridge's method, the answer was 4 - 4 =0, yet the booklet's answer was 4 - |- 4| = 8 units^2

So which one is right?

 

[ninetypercent]

Re: Don't quite understand a concept of integration - conflicting concepts - need hel

if you are told to integrate, then the answer is 0
if you are told to find the area, then the answer is 8 units^2

 

[ninetypercent]

Re: Don't quite understand a concept of integration - conflicting concepts - need hel

the reason why you have to split it up is because one part of the graph is below the x axis. If this occurs, you cannot simply calculate it by substituting the values because you could get a ZERO or negative area.

However, if you are simply told to evaluate the integral, then you can substitute the values in

 

[Mature Lamb]

Re: Don't quite understand a concept of integration - conflicting concepts - need hel

^ Yep. This is also why you should get a general idea of what the graph looks like if one isn't given already. e.g. x^3 from -2 to 2 would give you 0 if you just substitute values in.

 

[ohexploitable]

Re: Don't quite understand a concept of integration - conflicting concepts - need hel

The definite integral is the net area, in this case, areas below the x-axis are negative.

If you do physics think of it this way, integral is to area as displacement is to distance.

 

[blackops23]

Re: Don't quite understand a concept of integration - conflicting concepts - need hel

the reason why you have to split it up is because one part of the graph is below the x axis. If this occurs, you cannot simply calculate it by substituting the values because you could get a ZERO or negative area.

However, if you are simply told to evaluate the integral, then you can substitute the values in

You mean the end points of the interval right?

Thanks

 

[ohexploitable]

Re: Don't quite understand a concept of integration - conflicting concepts - need hel

You mean the end points of the interval right?

Thanks

Yes.

 

[blackops23]

Re: Don't quite understand a concept of integration - conflicting concepts - need hel

Hi, I could really use some help on this question: don't think I understood the wording of it or something:

Sketch y=x^2 and mark the points A(a,a^2), B(-a,a^2), P(a,0), Q(a,0)

(a) Show that a(integral sign)0 x^2 dx = 2/3 (area of triangle AOP)

Ok so the question's asking to find the area under y=x^2 between 0 and a, right? But I don't understand how that value is 2/3, I mean depending on the value of a, the area will change wouldn't it?

Anyways, I found a(integral sign)0 x^2 dx which equaled [a^3]/3. Then I calculated the area of triangle AOP which was a^3/2. I don't get what I'm supposed to do in this question, what exactly is it asking me, and why was (area of AOP) in the question? Because I definitely know that, a(integral sign)0 x^2 dx IS NOT EQUAL TO area of AOP.

Appreciate the help thanks guys.

 

[ohexploitable]

Re: Don't quite understand a concept of integration - conflicting concepts - need hel

 

[blackops23]

Re: Don't quite understand a concept of integration - conflicting concepts - need hel

lol crap, so I did misread the question, it meant (2/3)*(area of AOP). Silly silly me...