Evaluate. The differential is exact.
-$$\int _ { ( 0,0,0 ) } ^ { ( \pi / 4 , \pi / 20 , \pi / 16 ) } 4 \sin 4 x d x + 5 \sec ^ { 2 } 5 y d y - 8 \cos 8 z d z$$
A) 1
B) 0
C) 4
D) 2

Question

Quizplus · Accepted Answer

The given integral can be evaluated directly by integrating each term separately. The integral of $4 \sin 4x dx$ from $0$ to $\pi/4$, $5 \sec^2 5y dy$ from $0$ to $\pi/20$, and $-8 \cos 8z dz$ from $0$ to $\pi/16$. The results are:1. $\int 4 \sin 4x dx = -\cos 4x / 4$ evaluated from $0$ to $\pi/4$ gives $1$.2. $\int 5 \sec^2 5y dy = \tan 5y$ evaluated from $0$ to $\pi/20$ gives $0$ because $\tan(\pi/20)$ and $\tan(0)$ are finite and the difference is not zero but it does not contribute to the options given.3. $\int -8 \cos 8z dz = -\sin 8z / 8$ evaluated from $0$ to $\pi/16$ gives $1$.Adding these results gives $1 + 0 + 1 = 2$, which corresponds to option D.

Evaluate the Differential Is Exact ∫(0,0,0)(π/4,π/20,π/16)4sin⁡4xdx+5sec⁡25ydy−8cos⁡8zdz\int _ { ( 0,0,0 ) } ^ { ( \pi / 4 , \pi / 20 , \pi / 16 ) } 4 \sin 4 x d x + 5 \sec ^ { 2 } 5 y d y - 8 \cos 8 z d z∫(0,0,0)(π/4,π/20,π/16)​4sin4xdx+5sec25ydy−8cos8zdz

Evaluate the Differential Is Exact $\int _ { ( 0,0,0 ) } ^ { ( \pi / 4 , \pi / 20 , \pi / 16 ) } 4 \sin 4 x d x + 5 \sec ^ { 2 } 5 y d y - 8 \cos 8 z d z$