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JH
Numerade Educator

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Problem 12 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int_0^{\frac{\pi}{2}} (2 - \sin \theta)^2 d \theta $

Answer

$\frac{9 \pi}{4}-4$

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Video Transcript

This problem is from Chapter seven section to problem number twelve in the book Calculus Early Transcendental Sze, eighth Edition by James Store Here we have a definite integral from zero to a pie or two of parentheses to minus scientific data Square. So first thing we could do here is just evaluate this square and we have ah four minus for science data plus science. Where data Next, we can use an identity for Science Square. Richard, in a metric identity to rewrite this expression here, Sign square, and this becomes one minus coastline of two theater I'LL divided by two So here we could combine this four and one over two. Add those fractions together first to get it sign over too. Minus four. Signed data minus co. Sign of two data over to so we can evaluate each of these three girls. Oh, and it might help for this in a girl over here. The last general circled agreed to use the U substitution u equals today. So, evaluating the story we have a ninth eight over too. Plus for close on terra minus sign tooth data over two times two, which will give us a four and they're denominator. And we want to evaluate this new expression at zero and power too. So if we plug in pi over to first so co sign a pie or two zero and sign of two times pi over to a sign of pies also zero. So this is the first time we get after plugging in pi over too. And then when we plug in zero nine times here over to a zero four times coz I know zero is for and signer zeros also zero. So we're left with nine pi over four, minus four and that's our final answer.