Evaluate the integrals in Exercises 23–32.
∫₀^π √(1 - co...

Question

Evaluate the integrals in Exercises 23–32.

∫₀^π √(1 - cos²(θ)) dθ

Accepted Answer

Welcome back everyone. Evaluate the definite integral, integral from negative pi divided by 2 up 2 pi divided by 2 of the square root of 1 minus sin square of xDX. A says 0, B says 1, C says square root of 2, and D 2. For this problem, let's begin by remembering the Pythagorean identity sin square of x plus cosine squad of x is equal to 1. So if we subtract sin squared from both sides, we get 1 minus sin square x is equal to cosine squared of x. So in this problem we're integrating the square root of cosine squared of x, which is equal to the absolute value of cosine x. And now if we analyze the limits of integration. Notice that x is between. Negative pi divided by 2 and pi divided by 2, where cosine is non-negative, right? Specifically, we can say that in this interval. The absolute value of cosine of x is going to be equal to cosine of x itself. So now, having performed the analysis, we can say that we're integrating from pi divided by 2 up to pi divided by 2. Cosine x d x. This is now an elementary integral, and the integral of cosine x is sin x, and we want to evaluate sinx from pi divided by 2 up to pi divided by 2. Applying the fundamental theorem of calculus part 2, we get sin of pi divided by 2 minus sine of negative pi divided by 2, and we know that sine is an odd function, so that's sin of pi divided by 2. Minus Negative sign of pi divided by 2. Which is basically 2 sign off by divided by 2. And the value of sine of pi divided by 2 is 1, so we get 2 multiplied by 1, which is 2. And therefore the correct answer is option D. Thank you for watching.

Evaluate the integrals in Exercises 23–32.∫₀^π √(1 - cos²(θ)) dθ

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Evaluate the integrals in Exercises 23–32.
∫₀^π √(1 - cos²(θ)) dθ