Final answer:
To evaluate the given cylindrical coordinate integral π ∫0 1 ∫0 7-r2 ∫rdz r dr dθ, we can follow a step-by-step approach. First, evaluate the innermost integral with respect to z, then the middle integral with respect to r, and finally the outer integral with respect to θ. By substituting the results into the respective integral, we can evaluate the given integral.
Step-by-step explanation:
The given integral can be evaluated by using cylindrical coordinates. We will use the formula for evaluating triple integrals in cylindrical coordinates:
π ∫01 ∫07-r2 ∫r1 rdz r dr dθ
Step 1: Evaluate the innermost integral, which is with respect to z:
∫07-r2 rdz = r(7-r2)
Step 2: Evaluate the middle integral, which is with respect to r:
∫01 r(7-r2) dr = ∫01 7r-r3 dr = ½ r2-⅓ r4
Step 3: Evaluate the outer integral, which is with respect to θ:
∫0π ½ r2-⅓ r4 dθ = π(½ r2-⅓ r4)
Putting it all together, the evaluated integral is:
π(½ r2-⅓ r4)