Question

Will someone please check my answer for this question about integrals? Here is my work.

I am given:

x = y^3 - 3y
x = 5 - y^4

These are both solved for x, so they should equal each other:
y^3 - 3y = 5 - y^4
y^4 + y^3 - 3y - 5 = 0

I found the roots of the above equation to be -1.361918 and 1.55800365. I am thinking that the area of the enclosed area will be the integral from -1.361918 to 1.55800365.... but I don't know what the integrand will be. Answer C uses the same y^4 + y^3 - 3y - 5 equation, and Answer B uses something a little different. Other examples in my class suggesting that I need to do something like this:

(5 - y^4) - (y^3 - 3y)
(5 - y^4 - y^3 + 3y)
(-y^4 - y^3 + 3y + 5)

...and then let that answer be the integrand ^^ (which would reflect Answer B) But I'm not totally sure why I would subtract. Maybe someone could explain to me the difference between answers B and C and which one I should select?

Answer 1

the functions are in y-terms, so the integrand variable will be "y", and the red one is the right-function and the blue one is the left-function, so using

$\bf \displaystyle \int\limits_{a}^{b} [right-function]~~-~~[left~function]\qquad we~get \\\\\\ \displaystyle \int\limits_{\qquad -1.36192}^{\qquad 1.55800}~[(5-y^4)-(y^3-3y)]dy \\\\\\ \displaystyle \int\limits_{\qquad -1.36192}^{\qquad 1.55800}~(-y^4-y^3+3y+5)dy$