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

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

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

\int_{0}^{2 π} 1 - cos^{2} (x) d x

4

Solution steps

\int_{0}^{2 π} 1 - cos^{2} (x) d x

Rewrite using trig identities

= \int_{0}^{2 π} sin^{2} (x) d x

Apply exponent rule:

a^{2} = ∣ a ∣

= \int_{0}^{2 π} ∣ sin (x) ∣ d x

Eliminate Absolutes

= \int_{0}^{π} sin (x) d x + \int_{π}^{2 π} - sin (x) d x

\int_{0}^{π} sin (x) d x = 2

\int_{π}^{2 π} - sin (x) d x = 2

= 2 + 2

Simplify

= 4

\int_{0}^{1} 25 - x^{2} d x

\int_{1}^{9} \frac{9}{x} d x

\int_{- 2}^{2} 4 + y^{2} - 2 y^{2} d y

\int_{- 4}^{- 2 x + 2} \frac{1}{9} d y

\int_{- π}^{\frac{π}{2}} x cos (n x) d x

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