Video: Evaluating a Double Integral

Evaluate the double integral ∫_(0)^(𝜋) ∫_(0)^(𝑦) sin 𝑥 d𝑥 d𝑦.

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Video Transcript

Evaluate the double integral, the integral between 𝜋 and zero of the integral between 𝑦 and zero of sin 𝑥 d𝑥 d𝑦.

Double integrals are a way to integrate over a two-dimensional area. Among other things, they let us calculate the volume under a surface. As the name double integral suggests, we have to integrate twice.

In this specific question, we firstly need to integrate sin 𝑥 d𝑥 between the limits 𝑦 and zero. We will then integrate this answer with respect to 𝑦 between the limits 𝜋 and zero. Integrating sin 𝑥 with respect to 𝑥 gives us negative cos 𝑥. This is one of our standard integrals that we need to remember.

As always with definite integrals, we then need to substitute in the upper and lower limits and subtract our answers. Substituting in 𝑦 gives us negative cos 𝑦. Cos or cosine of zero is equal to one. This means that negative cos of zero is equal to negative one. We are left with negative cos 𝑦 minus negative one. 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.

We now need to integrate this expression with respect to 𝑦 between 𝜋 and zero. The integral of negative cos 𝑦 is negative sin 𝑦. And the integral of one is 𝑦. This means that the integral of negative cos 𝑦 plus one with respect to 𝑦 is negative sin 𝑦 plus 𝑦.

Once again, we need to substitute in our limits. Substituting in 𝜋 gives us negative of sin 𝜋 plus 𝜋. Substituting in the lower limit gives us negative sin of zero plus zero. The sin of 𝜋 is equal to zero. And the sin of zero is also equal to zero. This means that three of the four terms are equal to zero. And we are just left with positive 𝜋. The integral of negative cos 𝑦 plus one with respect to 𝑦 between 𝜋 and zero is equal to 𝜋.

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.

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