Evaluate. The differential is exact.
-$$\int _ { ( 0,0,0 ) } ^ { ( 1,1,1 ) } \left( 8 x - 10 x ^ { 9 } y ^ { 5 } \right) d x - 5 x ^ { 10 } y ^ { 4 } d y - 45 z ^ { 4 } d z$$
A) -6
B) -4
C) -5
D) 0

Question

Quizplus · Accepted Answer

We can check if the differential is exact by computing its partial derivatives and seeing if they are all equal. 
$$\frac{\partial}{\partial y}(8x-10x^9y^5)=-50x^9y^4$$
$$\frac{\partial}{\partial x}(-5x^{10}y^4)= -50x^9y^4$$
$$\frac{\partial}{\partial z}(-45z^4)=-180z^3$$
Since the partial derivatives are equal, the differential is exact. Therefore, we can use the first fundamental theorem of calculus:
$$\int _ { ( 0,0,0 ) } ^ { ( 1,1,1 ) } \left( 8 x - 10 x ^ { 9 } y ^ { 5 } \right) d x - 5 x ^ { 10 } y ^ { 4 } d y - 45 z ^ { 4 } d z = \Big[ 4x^2-2x^{10}y^5\Big]_0^1 - \Big[ x^{10}y^5 \Big]_0^1 - \Big[ 9z^5 \Big]_0^1$$
Plugging in the limits of integration, we get:
$$4-2-0-1+1-9 = -6$$
Therefore, the answer is A) -6.

Evaluate the Differential Is Exact ∫(0,0,0)(1,1,1)(8x−10x9y5)dx−5x10y4dy−45z4dz\int _ { ( 0,0,0 ) } ^ { ( 1,1,1 ) } \left( 8 x - 10 x ^ { 9 } y ^ { 5 } \right) d x - 5 x ^ { 10 } y ^ { 4 } d y - 45 z ^ { 4 } d z∫(0,0,0)(1,1,1)​(8x−10x9y5)dx−5x10y4dy−45z4dz

Evaluate the Differential Is Exact $\int _ { ( 0,0,0 ) } ^ { ( 1,1,1 ) } \left( 8 x - 10 x ^ { 9 } y ^ { 5 } \right) d x - 5 x ^ { 10 } y ^ { 4 } d y - 45 z ^ { 4 } d z$