Why is Fundamental Theorem of Calculus not applicable here?

[jamesfaucette]

See problem below.

"Let R be the region in the first quadrant bounded by the x-axis, the graph of x=(ky^2)+2 and the line x=4. Write an integral expression for the area of the region R and show that this area decreases as k increases."

I understand that there are multiple ways to set up the integral for the area of R. We could set up our integral in terms of either x or y. However, my question is, when we set it up in terms of y, can we use the first fundamental theorem of calculus to find the derivative of the area function? Of course, we could evaluate the integral using the second fundament theorem of calculus, and take the derivative of the result, which would give us the derivative of the area function, but I wanted to see if there was another way. I have attached work for reference.

Thanks!

 

[HallsofIvy]

You can take x from 0 to 4 and, for each x, y from 0 to \(\displaystyle \sqrt{\frac{x- 2}{k}}\).
The area is \(\displaystyle \int_0^4\int_0^{\sqrt{\frac{x-2}{k}}} dydx\).

To show that the area decreases as k increases, take the derivative with respect to k and show it is negative. You don't need the "Fundamental Theorem of Calculus". That derivative is \(\displaystyle -\int_0^4\int_0^{\sqrt{\frac{x-2}{k^2}}}dydx\). Is that negative?