quick antidifferentiation question?

Question

Evaluate this definite intergral
(2X+1)^3 dx between X=1 & X=0
Can you please show working out.
Also, what is the difference between a definate intergral & exact area?

Mat, the answer is 10, not 78.

Answer 1

∫(2x+1)³ dx,0,1
=|(2x+1)^4/(4X2)|,0,1
= 1/8 x (81-1)
= 10.

Answer 2

You are right, the answer is 10 units squared sorry.

Lets see if I can write it all out for you.

intergrate (2X+1)^3 between X=1 and X=0

First let u = 2X+1 so that the problem is now

intergrate u^3 between X=1 which is u = 3 (not 5 that was the error) and X=0 which is u =
1

next differentiate u with respect to x

u = 2X+ 1 so du/dx = 2

now the problem is

intergrate 0.5*(u^3) * (du/dx) * dx note that du/dx and the 0.5 factor when multiplied together give 1 so it does not change the equation at all by putting them in.

intergrate 0.5 (u^3) du as the two dx terms cancel

= 0.5 [ (u^4)/4] between u = 3 and u = 1

= 0.5[(3^4)/4 - (1^4)/4]

=0.5*[81/4 - 1/4]

= 0.5*80/4 = 10 units squared

And there is no difference between a definate intergral and exact area in a maths question.

Answer 3

∫(2x+1)³ dx,0,1=[{(2x+1)^4}/(4X2)],0,1
= 1/8 x (81-1)
= 10.

A definite integral is when you integrate a function with limits, just like this question. The definite integral between two points gives you the exact area under the curve.

Boy oh boy, Mat. You surely do show him from the basic. hehe. Anyway, it is good.

Answer 4

First, the concept "definite integral" exists in maths, but area is a concept existing in physics.

The latter must have units, while the formal is just numbers.

Thus, a definite integral can be negative, 0 or positive. However, an area must not be negative, and must have the appropriate unit corresponding to an area.

A definite integral w.r.t. x can only give you a number related to the area between the curve and the x-axis. Another step needs to transform this number to become the area.

---
hi doug_donaghue, yes, surely negative integrals are usually transformed into their corresponding physical meanings. I am saying physically, I don't know of any surface that has a negative area.

Answer 5

Yo!! back2nature.

You need to get back 2 the books. Negative area(s) most certainly *do* have physical meaning. If, for example, you get negative area on a thermal cycle diagram, it means that work is being done *on* the system rather than being done *by* the system.

There are also meanings for negative volumes


Doug

Answer 6

Okay, now I remembered why I hated math...