Although there are numerous ways to solve this problem, I reckon finding the area by integrating in respect to y would be the easiest method. In order to do so, you need to change the functions in terms of y and integrate it in respect to y. Here is how the calculations would look like.
y = x^2
x = +- sqrt(y)
As the horizontal. line y = k divides the area into two equal parts, we can set an equation stating that the area from y= 0 to y = k equals the area from y = k to y = 9. This is how it would look like:
[ (4/3) * (y)^(3/2) } from 0 to k = [ (4/3) * (y)^(3/2) } from k to 9
Definitely, it is possible to integrate in respect to x. I used the method of integrating in respect to y as I thought that this method is the most suitable one for this question.
Hi,
Thank you for your question.
Although there are numerous ways to solve this problem, I reckon finding the area by integrating in respect to y would be the easiest method. In order to do so, you need to change the functions in terms of y and integrate it in respect to y. Here is how the calculations would look like.
y = x^2
x = +- sqrt(y)
As the horizontal. line y = k divides the area into two equal parts, we can set an equation stating that the area from y= 0 to y = k equals the area from y = k to y = 9. This is how it would look like:
[ (4/3) * (y)^(3/2) } from 0 to k = [ (4/3) * (y)^(3/2) } from k to 9
(4/3) * (k)^(3/2) = (4/3) * (9)^(3/2) - (4/3) * (9)^(3/2)
(8/3) * (k)^(3/2) = 36
k = (27/2)^(2/3)
Therefore, the horizontal line would be y = (27/2)^(2/3).
Does this answer your question?