What is the integral from 0 to 2pi of sqrt(1-cos(θ)) ?

The integral from 0 to 2pi of sqrt(1-cos(θ)) is 4sqrt(2)

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integral from 0 to 2pi of sqrt(1-cos(θ))

\int_{0}^{2 π} 1 - cos (θ) d θ

42

Decimal Notation

5.65685 \dots

Solution steps

\int_{0}^{2 π} 1 - cos (θ) d θ

Rewrite using trig identities

= \int_{0}^{2 π} 2 sin^{2} (\frac{θ}{2}) d θ

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{0}^{2 π} sin^{2} (\frac{θ}{2}) d θ

Apply exponent rule:

a^{2} = ∣ a ∣

= 2 \cdot \int_{0}^{2 π} sin (\frac{θ}{2}) d θ

Eliminate Absolutes

= 2 \cdot \int_{0}^{2 π} sin (\frac{θ}{2}) d θ

Apply u-substitution

: \int_{0}^{π} sin (u) \cdot 2 d u

= 2 \cdot \int_{0}^{π} sin (u) \cdot 2 d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot 2 \cdot \int_{0}^{π} sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= 2 \cdot 2 [- cos (u)]_{0}^{π}

Compute the boundaries:

2

= 2 \cdot 2 \cdot 2

Simplify

= 42

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What is the integral from 0 to 2pi of sqrt(1-cos(θ)) ?
The integral from 0 to 2pi of sqrt(1-cos(θ)) is 4sqrt(2)