evaluate the integral 1 01dx4x2 Solution02 4x212 dx x 2sint

Solution

[0,2] (4-x^2)^1/2 dx
x = 2sint <-- make the substitution for x
dx = 2cost <--- make the substitution for dx
change the limits from [0,2] to [0,/6]

[0,/6] (4-(2sint)^2)^1/2 /2cost dt
[0,/6] (4-4sin²t))^1/2 /2cost dt
[0,/6] (4(1-sin²t))^1/2 /2cost dt
[0,/6] (4(cos²t))^1/2 /2cost dt
[0,/6] 2(cost) /2cost dt <-- from 0 to /2 sqrt(cos²t) is positive, so you can leave it as positive
[0,/6] dt
[0,/6] dt
[0,/6] t
/6 - 0
= /6-------> ANSWER