Solve Definite Integrals: F(x)=0.5x2-2 & F(x)=x3+x2-6x

[Sirsh]

Hi all, just wondering if someone could explain to me about define integrals.
Say i have F(x)=0.5x²-2 and F(x)=x³+x²-6x and i want to find the area of the regions which is satisfied from -2 to 2.

so $$\int$$0.5x²-2 from -2 to 2. $$\int$$x³+x²-6x from -2 to 2.

Now with the cubic, from -2 to 0 is supposedly positive and from 0 to 2 it's negative, is that true i thought the it doesn't matter which region the area is in it'll always be counted as positive.

so, $$\int$$x³+x²-6x from -2 to 0. = $$\int$$(-2)³+(-2)²-6(-2) - $$\int$$(0)³+(0)²-6(0) = 8-0 = 8units²


$$\int$$x³+x²-6x from 0 to 2. = $$\int$$(0)³+(0)²-6(0) - $$\int$$(2)³+(2)²-6(2) = 0- 0 = 0 units²

Therefore, Area from (-2 to 0)+ Area from (0 to 2) = 8+0 = 8units²

But using my calculator doing the same steps (except in the graphing section) doing -2 to 0 and adding it to 0 to 2's value i get 16units²( 10.667+5.333)

I don't know what's going on..

 

[Sirsh]

I only did the one area for the 2nd function, ignore the 1st function.

 

[pbandjay]

A definite integral is positive if the area is above the y-axis and negative if the area is below the y-axis. If you integrate over an interval that has both, the result will then not represent the complete physical area.

Looking at a graph of your cubic function: http://www.wolframalpha.com/input/?i=x^3%2Bx^2-6x

The area under the curve from -2 to 0 is larger than the area "above" the curve from 0 to 2. Since the positive area above (-2,0) is larger than the negative area below (0,2), the total area will be positive.

When I did these calculations, I got an area of 32/3 on (-2,0) and -16/3 on (0,2). So the definite integral on (-2,2) would be 32/3 - 16/3 = 16/3.

The total physical area would be 32/3 + 16/3 = 48/3 = 16. I had to use the absolute value of the areas to calculate the total physical area. I hope this clears things up!

 

[HallsofIvy]

Hi all, just wondering if someone could explain to me about define integrals.
Say i have F(x)=0.5x²-2 and F(x)=x³+x²-6x and i want to find the area of the regions which is satisfied from -2 to 2.

so $$\int$$0.5x²-2 from -2 to 2. $$\int$$x³+x²-6x from -2 to 2.

Now with the cubic, from -2 to 0 is supposedly positive and from 0 to 2 it's negative, is that true i thought the it doesn't matter which region the area is in it'll always be counted as positive.

so, $$\int$$x³+x²-6x from -2 to 0. = $$\int$$(-2)³+(-2)²-6(-2) - $$\int$$(0)³+(0)²-6(0) = 8-0 = 8units²

You haven't integrated anything, you have just evaluated the function itself at -2 and 0 and subtracted. The integral is
$$\int_{-2}^0 x^3+ x^2- 6x dx= \left[\frac{1}{4}x^4+ \frac{1}{3}x^3- 3x^2\right]_{-2}^0$$
$$= 0- \left(\frac{1}{4}(16)+ \frac{1}{3}(-8)- 3(4)\right)= \frac{32}{3}$$

$$\int$$x³+x²-6x from 0 to 2. = $$\int$$(0)³+(0)²-6(0) - $$\int$$(2)³+(2)²-6(2) = 0- 0 = 0 units²

Therefore, Area from (-2 to 0)+ Area from (0 to 2) = 8+0 = 8units²

But using my calculator doing the same steps (except in the graphing section) doing -2 to 0 and adding it to 0 to 2's value i get 16units²( 10.667+5.333)

I don't know what's going on..

That's because your calculator integrated and you didn't!

 

[Sirsh]

With both did you do F(b) - F(a) ? also, do you know anything about these type of intergrals.

$$\int$$f(x)dx+|$$\int$$f(x)dx|
|$$\int$$f(x)dx|+$$\int$$f(x)dx
|$$\int$$f(x)dx+$$\int$$f(x)dx|

I'm unsure what these | | parts do when theyre associated with the integral. thanks a lot.

 

[HallsofIvy]

With both did you do F(b) - F(a) ? also, do you know anything about these type of intergrals.

$$\int$$f(x)dx+|$$\int$$f(x)dx|
|$$\int$$f(x)dx|+$$\int$$f(x)dx
|$$\int$$f(x)dx+$$\int$$f(x)dx|

I'm unsure what these | | parts do when theyre associated with the integral. thanks a lot.

$$\left|\int f(x)dx\right|$$
means "first take the integral of the function then take the absolute value of that number".

$$\int |f(x)|dx$$
means "first take the absolute value of the function, then integrate".

 

[Sirsh]

Wow, I'm so retarded I forgot to actually intergrate the function. how embarrassing..

 

[Sirsh]

$$\left|\int f(x)dx\right|$$
means "first take the integral of the function then take the absolute value of that number".

$$\int |f(x)|dx$$
means "first take the absolute value of the function, then integrate".

Thanks a lot HallsofIvy, I don't really understand. could you please do an example of both?