Evaluate the following iterated integral.
$\int_{0}^{6}\int_{0} ^ {4}\int_{0} ^ {2}(-4 x+5 y+z) d x d y d z $
A) 428
B) 438
C) 431
D) 442
E) 432

Question

Evaluate the following iterated integral. 
$\int_{0}^{6}\int_{0} ^ {4}\int_{0} ^ {2}(-4 x+5 y+z) d x d y d z $
A) 428
B) 438
C) 431
D) 442
E) 432

Quizplus · Accepted Answer

We can evaluate the integral by integrating with respect to one variable at a time.

First, we integrate with respect to x:

$$\int_{0}^{6}\int_{0} ^ {4}\int_{0} ^ {2}(-4 x+5 y+z) d x d y d z = \int_{0} ^ {6}\int_{0} ^ {4}\left[-2x^2+5xy+xz\right]_{0} ^ {2}d y d z$$

Simplifying the integral, we get:

$$\int_{0} ^ {6}\int_{0} ^ {4}(8-4y+2z) d y d z = \int_{0} ^ {6}\left[8y-2y^2+2yz\right]_{0} ^ {4}d z$$

Simplifying further, we get:

$$\int_{0} ^ {6}(32-32z) d z = \left[32z-16z^2\right]_{0} ^ {6} = 432$$

Therefore, the answer is E) 432.

Evaluate the Following Iterated Integral ∫06∫04∫02(−4x+5y+z)dxdydz\int_{0}^{6}\int_{0} ^ {4}\int_{0} ^ {2}(-4 x+5 y+z) d x d y d z ∫06​∫04​∫02​(−4x+5y+z)dxdydz