We want to evaluate
the given double integral, but before we do,
let's look at the graph of the integrand function
over the rectangular region of integration
given by the limits of integration.
The integrand function is graphed in blue,
and the region of integration in the xy-plane
is graphed in yellow.
Notice how the blue surface is above and below the xy-plane
over the region of integration,
which means the double integral
does not represent the volume bounded by the surface
in the xy-plane.
We can still evaluate the double integral,
but it does not represent volume.
Looking at the double integral,
we integrate with respect to x first.
And because we have a factor of x here,
and we have sine of the quantity x squared plus y squared,
we'll have to perform u substitution.
We'll let u be equal to x squared plus y squared.
And therefore differential u is going to be equal
to the derivative of x squared plus y squared
with respect to x times d x,
which would be two x d x.
Our integrand function does not have a two,
so let's go ahead and divide both sides by two here.
So now we know that 1/2 differential u
is equal to x d x.
So if we write all of this in terms of u,
again x d x is equal to 1/2 d u.
So we have 1/2 d u,
and then we'd have a factor of y.
And then we'd have sine of x squared plus y squared
is equal to u.
The antiderivative of 1/2 y sine u with respect to u
would be negative 1/2 y cosine u,
which means the antiderivative with respect to x
would be negative 1/2 y times cosine of u,
which is really the quantity x squared plus y squared.
And now we need to find big f of b minus big f of a.
Let's go ahead and write this as
the integral from zero to the square root
of two pi divided by three.
We're making a substitution here for x,
so let's write this as negative 1/2 y times,
when x is equal to the square root
of seven pi divided by six,
we'd have cosine of
seven pi divided by six plus y squared
minus, when x is zero we would just have cosine y squared.
And now we need to integrate with respect to y.
That is also going to require u substitution.
Let's do this on the next slide.
Here we'll let u equal
seven pi divided by six plus y squared.
Notice how here we just have cosine y squared,
so let's also solve this for y squared.
Notice that y squared
would be equal to u minus seven pi divided by six.
And now we need to find differential u,
which is the derivative of seven pi divided by six
plus y squared, with respect to y,
times differential y, which would be two y d y.
And again, we don't have a factor of two,
so let's go ahead and divide both sides by two.
So we know that 1/2 differential u
is equal to y d y.
So writing all of this in terms of u,
we have this factor of negative 1/2.
And then y d y is equal to 1/2 d u,
so we have another factor of 1/2.
And then we're going to have cosine
of seven pi divided by six plus y squared is equal to u.
And then we have minus cosine y squared,
where y squared is going to be equal
to u minus seven pi divided by six.
So notice how the antiderivative with respect to u
would be equal to negative 1/4 sine u
minus sine of the quantity u minus seven pi divided by six,
which means, with respect to y,
the antiderivative is going to be
negative 1/4 times, again we'll have sine u,
which is really sine of seven pi
divided by six plus y squared
minus,
and then the sine of the quantity
u minus seven pi divided by six
would really just be sine of y squared.
And now performing substitution for y,
we're going to have negative 1/4 times the quantity,
well, when y is equal to the square root
of two pi divided by three.
Here we're going to have sine of seven pi divided by six
plus two pi divided by three
minus sine of y squared, would just be two divided by three.
And then we have minus, when y is zero,
we'd have sine of seven pi divided by six
minus sine zero.
Well, seven pi divided by six plus two pi divided by three
would be seven pi divided by six
plus four pi divided by six.
So this angle here is 11 pi divided by six.
The sine of 11 pi divided by six
is equal to negative 1/2,
and then we have minus the sine of two pi divided by three.
The sine of two pi divided by three
is square root three divided by two.
And then we have minus
sine seven pi divided by six is equal to negative 1/2.
And sine of zero is zero.
So notice here we have negative 1/2
and then minus negative 1/2,
which becomes negative 1/2 plus 1/2.
So these two terms simplify to zero.
So we're left with negative 1/4
times negative square root three divided by two,
which gives an exact value
of positive square root three divided by eight,
or as a decimal approximation,
it's approximately 0.2165.
I hope you found this helpful.