Evaluate the integral upper limit pi and lower limit 7pi/2 of (1-cos2tdt)/2
To evaluate the integral ∫[7π/2, π] (1 - cos(2t)) dt/2, we can follow these steps:
Step 1: Simplify the integrand.
The numerator can be simplified using the double-angle formula for cosine:
1 - cos(2t) = 1 - (cos²(t) - sin²(t))
= 1 - cos²(t) + sin²(t)
= 1 - (1 - sin²(t)) + sin²(t)
= 2sin²(t)
So, the integrand becomes (2sin²(t))/2, which simplifies to sin²(t).
Step 2: Evaluate the integral.
Now, we have ∫[7π/2, π] sin²(t) dt.
To solve this integral, we can use the trigonometric identity for sin²(t):
sin²(t) = (1 - cos(2t))/2
Therefore, the integral simplifies to:
∫[7π/2, π] (1 - cos(2t))/2 dt
Step 3: Apply the limits.
Now, we can apply the limits of integration to the integral.
For the lower limit, t = 7π/2:
∫[7π/2, π] (1 - cos(2t))/2 dt
= ∫[7π/2, π] (1 - cos(2(7π/2)))/2 dt
= ∫[7π/2, π] (1 - cos(7π))/2 dt
= ∫[7π/2, π] (1 - cos(3π))/2 dt
= ∫[7π/2, π] (1 - cos(π))/2 dt
= ∫[7π/2, π] (1 + 1)/2 dt
= ∫[7π/2, π] dt
= (π - 7π/2)
For the upper limit, t = π:
∫[7π/2, π] (1 - cos(2t))/2 dt
= ∫[7π/2, π] (1 - cos(2π))/2 dt
= ∫[7π/2, π] (1 - cos(0))/2 dt
= ∫[7π/2, π] (1 - 1)/2 dt
= 0
Step 4: Final result.
To find the overall result, we subtract the result at the lower limit from the result at the upper limit:
(π - 7π/2) - 0 = -6π/2 - π = -7π/2 - π = -9π/2.
Therefore, the value of the integral ∫[7π/2, π] (1 - cos(2t))/2 dt is -9π/2.