Use the Evaluation Theorem to find the exact value of the integral
�ç from 1/2 to 0 (a/�ã(1−x^2)dx.
The answer should involve the parameter a.
To find the exact value of the integral ∫(1/2 to 0) [(a/√(1−x^2))]dx using the Evaluation Theorem, we need to follow these steps:
Step 1: Find the antiderivative F(x) of the integrand.
Step 2: Evaluate F(x) at the upper limit of integration (1/2).
Step 3: Evaluate F(x) at the lower limit of integration (0).
Step 4: Subtract the result in Step 3 from the result in Step 2.
Step 1: Find the antiderivative F(x) of the integrand.
The integral ∫(a/√(1−x^2)) dx can be found by using the substitution u = 1−x^2.
Differentiating both sides with respect to x, we get du/dx = −2x. Solving for dx, we have dx = du/−2x.
Substituting these values into the integral:
∫(a/√(1−x^2)) dx = ∫(a/√u) (du/−2x)
Since x = √(1−u), we can substitute this back into the integral:
∫(a/√(1−x^2)) dx = ∫(a/√u) (du/−2√(1−u)).
Simplifying, we have:
∫(a/√(1−x^2)) dx = (−a/2) ∫(du/√u).
Now, integrating ∫(du/√u) gives:
∫(a/√(1−x^2)) dx = (−a/2) * 2√u + C.
Step 2: Evaluate F(x) at the upper limit of integration (1/2).
Substituting u = 1−(1/2)^2 = 1−1/4 = 3/4 into the antiderivative, we get:
F(1/2) = (-a/2) * 2√(3/4) + C.
Step 3: Evaluate F(x) at the lower limit of integration (0).
Substituting u = 1−0^2 = 1 into the antiderivative, we get:
F(0) = (-a/2) * 2√1 + C.
Step 4: Subtract the result in Step 3 from the result in Step 2.
To find the exact value of the integral, we subtract F(0) from F(1/2):
∫(1/2 to 0) [(a/√(1−x^2))]dx = F(1/2) - F(0).
= (-a/2) * 2√(3/4) + C - (-a/2) * 2√1 + C.
Simplifying further, we have:
∫(1/2 to 0) [(a/√(1−x^2))]dx = -a√3 + a.
Therefore, the exact value of the integral is given by:
∫(1/2 to 0) [(a/√(1−x^2))]dx = a - a√3.