Find the area enclosed by the following functions from x=0 to x=2:

Question

Find the area enclosed by the following functions from x=0 to x=2:

y=sqrt*(x+2)
y=1/(1+x)

I got
2/3(8)-ln3-2/3(sqrt*8)
can someone please verify this?

Answer 1

2/3(8)-ln3-2/3(sqrt*8)

My answer,
16/3 - ln 3 - 4/3(sqrt(2))

You have 16/3,
2/3 (8) = 16/3

Where you went wrong,
You have, 2/3 (sqrt(8))
It should be,
2/3 (0 + 2)^3/2
2/3 (2)^3/2
2/3 (sqrt(2))^3
2/3 (2(sqrt(2)))
4/3 (sqrt(2))

Answer 2

Perfect, thanks!

I agree that your answer is more simplified..
but my answer is definitely not wrong because
2/3(sqrt(8)) and 4/3(sqrt(2)) are the same.

Because.. 8=4x2 and sqrt of 4 is 2.. so you take 2 out of the sqrt to be 2. 2x2 is 4. so you end up with 4/3(sqrt(2)). Do you agree?

Answer 3

I guess it depends on your teacher, because

(2)^3/2 means (sqrt(2))^3

I've never seen anyone interpret this as
(sqrt(2^3)).

The first way will always get you the correct, simplified answer. If I were you, I would get in the habit of doing this the first way.

Radicals should always be in the lowest terms. Any teacher I had would take off half credit for your answer.

Good luck.

Answer 4

To find the area enclosed by the given functions, you need to find the area between the two curves. This can be done by subtracting the area under the curve of the function with the smaller y-values from the area under the curve of the function with the larger y-values.

First, let's find the intersection points of the two curves by setting them equal to each other:

sqrt(x+2) = 1/(1+x)

To solve this equation, we need to square both sides:

x + 2 = 1/(1+x)^2

Now we can simplify the equation:

(1+x)^2 = 1/(x+2)
(x+1)^2 = 1/(x+2)

Take the square root of both sides:

x+1 = ±sqrt(1/(x+2))

Remove the square root from the denominator by squaring both sides again:

(x+1)^2 = 1/(x+2)

Expand the square on the left side:

x^2 + 2x + 1 = 1/(x+2)

Multiply both sides by (x+2) to remove the fraction:

(x+2)(x^2 + 2x + 1) = 1

Expand and simplify:

x^3 + 4x^2 + 5x + 2 = 1

x^3 + 4x^2 + 5x + 1 = 0

Now you can solve this polynomial equation to find the intersection points. However, it seems that solving this equation analytically might be challenging. You can try using numerical methods or graphing both functions to estimate the intersection points.

Once you have the intersection points, you can integrate the difference between the two functions over the interval [0, 2] to find the area enclosed.

Unfortunately, without the intersection points, it is not possible to verify the result you provided.