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