Evaluate. The differential is exact.
-$$\int _ { ( 0,0,0 ) } ^ { ( 4,6,2 ) } \left( 2 x y ^ { 2 } - 2 x z ^ { 2 } \right) d x + 2 x ^ { 2 } y d y - 2 x ^ { 2 } z d z$$
A) 512
B) 0
C) 1024
D) 640

Question

Quizplus · Accepted Answer

The given differential is exact, meaning it is the differential of some potential function $ f(x, y, z) $. By integrating the components, we find $ f(x, y, z) = x^2 y^2 - x^2 z^2 $. Evaluating $ f $ at the endpoints $(4, 6, 2)$ and $(0, 0, 0)$, we get $ f(4, 6, 2) = 512 $ and $ f(0, 0, 0) = 0 $. Thus, the integral evaluates to $ 512 - 0 = 512 $.

Evaluate the Differential Is Exact ∫(0,0,0)(4,6,2)(2xy2−2xz2)dx+2x2ydy−2x2zdz\int _ { ( 0,0,0 ) } ^ { ( 4,6,2 ) } \left( 2 x y ^ { 2 } - 2 x z ^ { 2 } \right) d x + 2 x ^ { 2 } y d y - 2 x ^ { 2 } z d z∫(0,0,0)(4,6,2)​(2xy2−2xz2)dx+2x2ydy−2x2zdz

Evaluate the Differential Is Exact $\int _ { ( 0,0,0 ) } ^ { ( 4,6,2 ) } \left( 2 x y ^ { 2 } - 2 x z ^ { 2 } \right) d x + 2 x ^ { 2 } y d y - 2 x ^ { 2 } z d z$