I'm reading this as

with

.
The value of the integral will be independent of the path if we can find a function

that satisfies the gradient equation above.
You have

Integrate

with respect to

. You get


Differentiate with respect to

. You get
![\dfrac{\partial f}{\partial y}=\dfrac{\partial}{\partial y}[x^2e^{-y}+g(y)]](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3D%5Cdfrac%7B%5Cpartial%7D%7B%5Cpartial%20y%7D%5Bx%5E2e%5E%7B-y%7D%2Bg%28y%29%5D)


Integrate both sides with respect to

to arrive at



So you have

The gradient is continuous for all

, so the fundamental theorem of calculus applies, and so the value of the integral, regardless of the path taken, is
Rewrite as (2^1/2 + 2^1/2)*(18^1/2-2^1/2) = 8
Answer:
john drove for 3 hours at a rate of 50 miles per hour and for 2 hours at 60 miles per hour.
Step-by-step explanation:
1500m=1.5km so 1500 divided by 10= 150s it takes 150s to cover a distance of 1.5km
First aquarium dimensions:
Length = 6 m.
Width = 4 m and
Height = 2 meter.
Second aquarium dimensions:
Length = 8 m.
Width = 9 m and
Height = 3 meter.
We know formula for volume of a cuboidal box = Length*Width*Height.
Plugging values of length, width and height of first aquarium in formula of volume. We get
V1 = 6*4*2 = 48 m^3.
Plugging values of length, width and height of second aquarium in formula of volume. We get
V2 = 8*9*3 = 216 m^3.
In order to find the total cubic meters of space do the sea turtles have in their habitat, we need to add both volumes.
Therefore, Total voulme of both aquarium = V1 +V2 = 48+216 = 264 m^3.
Therefore, total 264 m^3 cubic meters of space the sea turtles have in their habitat.