Question

In Exercises 1-4, evaluate the iterated integral. $$ \int_{0}^{4} \int_{0}^{\pi / 2} \int_{0}^{2} r \cos \theta d r d \theta d z $$

Short answer

The result of the triple integral is 8.

Step-by-step solution

Evaluate the innermost integral (integration over \( r \))

The result of the inner integral \( \int_{0}^{2} r \cos \theta d r \) can be computed by using the power rule for integration. This rule states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \). Here n = 1. Therefore, the inner integral evaluates to \( \frac{r^2}{2} \cos \theta \), evaluated between 0 and 2. Thus, the result of the innermost integral is \( 2 \cos \theta \).

Evaluate the intermediate integral (integration over \( \theta \))

The integral obtained from Step 1 is then substituted into the intermediate integral, yielding \( \int_{0}^{\pi / 2} 2 \cos \theta d \theta \). The antiderivative of \( \cos \theta \) is \( \sin \theta \), evaluated between 0 and \( \pi / 2 \), which results in 2.

Evaluate the outermost integral (integration over \( z \))

Lastly, the constant value 2 obtained from step 2 is substituted into the outermost integral, yielding \( \int_{0}^{4} 2 d z \), which can be evaluated simply by multiplying 2 by the length of the interval from 0 to 4, resulting in 8.