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Evaluate the double integral, the integral between 𝜋 and zero of the integral between 𝑦 and zero of sin 𝑥 d𝑥 d𝑦.
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Double integrals are a way to integrate over a two-dimensional area.
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Among other things, they let us calculate the volume under a surface.
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As the name double integral suggests, we have to integrate twice.
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In this specific question, we firstly need to integrate sin 𝑥 d𝑥 between the limits 𝑦 and zero.
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We will then integrate this answer with respect to 𝑦 between the limits 𝜋 and zero.
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Integrating sin 𝑥 with respect to 𝑥 gives us negative cos 𝑥.
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This is one of our standard integrals that we need to remember.
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As always with definite integrals, we then need to substitute in the upper and lower limits and subtract our answers.
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Substituting in 𝑦 gives us negative cos 𝑦.
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Cos or cosine of zero is equal to one.
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This means that negative cos of zero is equal to negative one.
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We are left with negative cos 𝑦 minus negative one.
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As the two negatives turn into a positive, our answer for the integral of sin 𝑥 d𝑥 between 𝑦 and zero is negative cos 𝑦 plus one.
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We now need to integrate this expression with respect to 𝑦 between 𝜋 and zero.
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The integral of negative cos 𝑦 is negative sin 𝑦.
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And the integral of one is 𝑦.
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This means that the integral of negative cos 𝑦 plus one with respect to 𝑦 is negative sin 𝑦 plus 𝑦.
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Once again, we need to substitute in our limits.
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Substituting in 𝜋 gives us negative of sin 𝜋 plus 𝜋.
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Substituting in the lower limit gives us negative sin of zero plus zero.
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The sin of 𝜋 is equal to zero.
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And the sin of zero is also equal to zero.
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This means that three of the four terms are equal to zero.
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And we are just left with positive 𝜋.
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The integral of negative cos 𝑦 plus one with respect to 𝑦 between 𝜋 and zero is equal to 𝜋.
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We can therefore say that the double integral, the integral between 𝜋 and zero of the integral between 𝑦 and zero of sin 𝑥 d𝑥 d𝑦 is equal to 𝜋.