Evaluate the Integral

Problem

(∫_0^8)(2*π*(0.5*x√(,1+(0.5)2))*d(x))

Solution

1. Simplify the constant terms inside the integral.

1+(0.5)2=1+0.25=1.25

2. Calculate the square root of the constant.

√(,1.25)=√(,5/4)=√(,5)/2

3. Combine all constant factors outside the integral.

2*π⋅0.5⋅√(,5)/2=2*π⋅1/2⋅√(,5)/2=(π√(,5))/2

4. Rewrite the integral with the constants moved outside.

(π√(,5))/2*(∫_0^8)(x*d(x))

5. Apply the power rule for integration to the function x

(∫_^)(x*d(x))=(x2)/2

6. Evaluate the definite integral at the boundaries 0 and 8

[(x2)/2]80=8/2−0/2

64/2=32

7. Multiply the result by the constant factor.

(π√(,5))/2⋅32=16*π√(,5)

Final Answer

(∫_0^8)(2*π*(0.5*x√(,1+(0.5)2))*d(x))=16*π√(,5)

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