Evaluate the Integral

Problem

(∫_0^1)(b√(,1−b2)*d(b))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function 1−b2 is proportional to the outer factor b

2. Define the substitution variable u=1−b2

3. Calculate the differential d(u) by differentiating u with respect to b which gives d(u)=−2*b*d(b) or −1/2*d(u)=b*d(b)

4. Determine the new limits of integration by substituting the original bounds into the equation for u when b=0 u=1 when b=1 u=0

5. Substitute the variables and limits into the integral to rewrite it in terms of u

(∫_1^0)(√(,u)*(−1/2)*d(u))

6. Simplify the integral by reversing the limits and removing the negative sign.

1/2*(∫_0^1)(u(1/2)*d(u))

7. Integrate using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)

1/2*[(u(3/2))/(3/2)]10

8. Evaluate the expression at the upper and lower bounds.

1/2⋅2/3*[1(3/2)−0(3/2)]

9. Multiply the constants to find the final value.

1/3⋅1=1/3

Final Answer

(∫_0^1)(b√(,1−b2)*d(b))=1/3

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