Evaluate the Integral

Problem

(∫_−2^1)(√(,3−x2)*d(x))

Solution

1. Identify the integral as the area under a circular arc. The integrand y=√(,3−x2) represents the upper half of a circle centered at the origin with radius r=3

2. Apply the trigonometric substitution x=3*sin(θ) to transform the integral. The differential is d(x)=3*cos(θ)*d(θ)

3. Determine the new limits of integration. When x=−2 θ=arcsin(−2/3) When x=1 θ=arcsin(1/3)

4. Substitute the variables into the integral. The expression √(,3−(3*sin(θ))2) simplifies to 3*cos(θ)

5. Simplify the resulting integral using the identity cos2(θ)=(1+cos(2*θ))/2

(∫_^)(9*cos2(θ)*d(θ))=9/2*(∫_^)((1+cos(2*θ))*d(θ))

6. Integrate the expression with respect to θ

9/2*(θ+sin(2*θ)/2)=9/2*(θ+sin(θ)*cos(θ))

7. Revert to the variable x using the relationships sin(θ)=x/3 and cos(θ)=√(,9−x2)/3

9/2*arcsin(x/3)+(x√(,9−x2))/2

8. Evaluate the definite integral at the boundaries x=1 and x=−2

(9/2*arcsin(1/3)+(1√(,8))/2)−(9/2*arcsin(−2/3)+(−2√(,5))/2)

9. Combine the terms to find the final numerical value.

9/2*arcsin(1/3)+√(,2)+9/2*arcsin(2/3)+√(,5)

Final Answer

(∫_−2^1)(√(,3−x2)*d(x))=9/2*(arcsin(1/3)+arcsin(2/3))+√(,2)+√(,5)

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