Evaluate∫ π20∫ a cos θ0r4 dr dθ

To evaluate the given integral, we will use the method of double integration.

Step 1: Evaluate the inner integral with respect to r: ∫ a cos θ0r^4 dr

To integrate this expression, we will use the power rule for integration. Applying the power rule, we get: = (a cos θ0) * (r^5/5) + C1

Step 2: Evaluate the outer integral with respect to θ: ∫ π20 [(a cos θ0) * (r^5/5) + C1] dθ

To integrate this expression, we will treat (a cos θ0) * (r^5/5) + C1 as a constant with respect to θ. Applying the integral of a constant, we get: = [(a cos θ0) * (r^5/5) + C1] * θ + C2

Step 3: Evaluate the definite integral: ∫ π20 [(a cos θ0) * (r^5/5) + C1] dθ

To evaluate the definite integral, we substitute the upper limit (20) and the lower limit (π) into the expression: = [(a cos θ0) * (r^5/5) + C1] * 20 - [(a cos θ0) * (r^5/5) + C1] * π

Simplifying further, we get the final result: = 20(a cos θ0) * (r^5/5) - π(a cos θ0) * (r^5/5) + 20C1 - πC1

Therefore, the evaluated integral is 20(a cos θ0) * (r^5/5) - π(a cos θ0) * (r^5/5) + 20C1 - πC1.