Question

Find the area of the shaded region. (Round your answer to two decimal places.)

Answer 1

@jim_thompson5910

Answer 2

@amistre64

Answer 3

@electrokid

Answer 4

notice that you are asked to find the shared AREA
if we integrate over the entire interval, we might get "0" as the answer because the area on the left side of "y-axis" will become negative.
But we can explaoit the symmetry and say that
\[A=2\times \int _{x=0}^2ydx\]

Answer 5

That seems a little more complex than what I need here

Answer 6

\[
A=2\int _0^2{2x\over x^2+4}dx\\
let\quad u=x^2+4\implies du=2xdx\\
{\rm when}\qquad x=0, u=4\\
{\rm when}\qquad x=2, u=8\\
A=2\int_{u=4}^8{1\over u}du\\
A=2\times\left[\ln|u|\right]_4^8\\
A=2\left(\ln8-\ln4\right)\\
A=2\ln\left(8\over4\right)\\
\Large\boxed{A=2\ln2}
\]

Answer 7

That is not the correct answer

Answer 8

two decimal places..
did you put the answer in decimals?

Answer 9

my apologies I did not convert to decimal. you are correct I am really sorry

Answer 10

good.
in the first paragraph, I saw the graph and realised that a direct integration will not give me the answer.
I have to split the integration into two halves. and then continue.

Answer 11

did you follow? including the substitution I did while evaluating the integration

Answer 12

No, I don't think I really followed. I don't know why you changed the integration from -2 to 2 to the integration from 0 to 2 with a -2 on the outside of the integral

Answer 13

the region on the left side of y-axis will have a negative area

Answer 14

because it lies below the X-axis.

Answer 15

However, I do understand that you cannot just take the integral directly due to the answer of "0" being obtain

Answer 16

yes. this should give a hint that somewehre the area is getting subtracted!!

Answer 17

Area in geometry is simply a magnitude of number.
interpretation of integration leads to a concept of "negative area" which tells us if the area lies "above" or "bel

Answer 18

he x-axis

Answer 19

and since the function "y" is symmetric,
I can say that the area under the curve from 0 to 2
equals
the area under the curve from 0 to -2

hence I split it that way

Answer 20

kapeesh?

Answer 21

um sort of. let me process it