Question

area under the curve

Answer 1

please let me know if I've represented area vs definite integral of the function between the interval [-2,1]

Answer 2

area between x-axis and curve in [-2,1]
Required area=\[\int\limits_{-2}^{-1} -f(x) dx+\int\limits_{-1}^{1} f(x)dx\]

Answer 3

Do you want to know if your answer is correct or if you've shown the work correctly?

Answer 4

-f(x) for symmetry @surjithayer ?

Answer 5

Is that what you're looking for? Area between curve and x-axis, or just area under the curve (which can include negative areas).
If you want absolute areas, you can do like surji suggests or just take absolute value of definite integrals, but you still need to separate intervals between where function equals zero.

Answer 6

(The -f(x) is not for symmetry, it is to have an equivalent effect of taking absolute value.)

Answer 7

between the curve and x-axis from x=-2 to x=1

Answer 8

\[A=\left|\int_{-2}^{-1}x^3+x^2dx\right|+\left|\int_{-1}^1x^3+x^2dx\right| \]

Answer 9

right by making -y we yield the same equation but instead of being under the x-axis it now be above the x-axis. that is how I understood it. but I also see the absolute value

Answer 10

Yes, means the same thing. I prefer to use absolute value because I know areas are always positive values and I don't want to make a rookie mistake of not distributing a negative sign correctly.

Answer 11

I agree with you, and I utilize the same and it leaves me fewer work such as identifying which function is greater when solving for the area of different intersecting continuous functions

Answer 12

it appears that I've represented the area and definite integral appropriately
thank you, guys

Answer 13

The area between two functions is: \[
\int_a^b|f(x)-g(x)|\;dx
\]In this case \(g(x)=0\). So it is just \[
\int_a^b|f(x)|\;dx
\]