The zoo is building a new polar bear exhibit,
and wants to put a semi-circular window in the concrete wall
of the swimming tank, as we see pictured here.
If the semi-circle has diameter 70 centimeters
and the bottom of the window is at a depth
of two point seven five meters, we want to find
the hydrostatic force on the window.
So the bottom of the window is at a depth
of two point seven five meters.
So this distance here is two point seven five meters.
We need to be careful here because the diameter is given
in centimeters, so if we put this semi-circle
on the coordinate plane,
the diameter would be point seven meters
so we have common units and therefore
if the center is at the origin,
on the right we'd have point three five meters,
on the left we'd have negative point three five meters.
Since the center is at the origin,
we could right the equation of the semi-circle
as x squared plus y squared
equals the radius squared, which would be
point three five squared,
as long as you restrict the value of y
so that it's greater than or equal to zero.
And now we'll find the hydrostatic force
by finding this integral here
where the force is equal to the integral of w times d of y
time L of y, integrated with respect to y from c to d,
where w is the weight density,
which is equal to the density of the water
times the acceleration due to gravity.
Now some problems we may be given the weight density
but in this problem we'll have to calculate it.
d of y is the depth of the fluid at y
and L of y is the horizontal length at y.
So let's set up our integral on the next slide.
Let's begin by determining the weight density,
which again is equal to the density of the water
times the acceleration due to gravity,
which would be 1,000 kilograms per meters cubed,
which is given
times nine point eight meters per second squared,
which is also given.
And therefore, the weight density would be 9,800
and this would Newtons per meters cubed.
Now before we find d of y and L of y,
it's helpful to sketch a horizontal slice of the window
using a rectangle, let's say, this rectangle here.
Notice how the height of this rectangle would be delta y,
which is y we integrate with respect to y.
And now if we let the length from the rectangle
to the origin be equal to y,
d of y would be the length from this rectangle
to the top of the water.
So if this represents our this slice or rectangle
and this length here is y,
notice how the depth would be two point seven five meters
minus y.
So this length here, d of y,
equals two point seven five minus y.
And now we need to find L of y,
the horizontal width of this slice.
Well we know the equation of our semicircle
is x squared plus y squared equals point three five squared
as long as y is restricted to being greater than
or equal to zero
and therefore the distance from the y axis
to the right edge would be x units
and so the distance from the y-axis
to the left edge here.
And therefore the total horizontal length of the window
and any value of y would be 2x.
But we need to have L as a function of y
not as a function of x.
So now we'll solve our equation for x
and then make a substitution here.
So if we know that x squared plus y squared
equals point three five squared,
again we'll solve this for x.
So we can write L as a function of y,
so we have x squared equals point three five squared
minus y squared,
and now we'll square-root both sides of the equation.
And because our x is a length, we only have
to consider the positive square root.
So we have x equals the square root
of point three five squared minus y squared.
And therefore L of y is equal to two times
the square root of point three five squared minus y squared.
And now we can set up the integral.
We know w, we know d of y,
and now we know L of y.
The hydrostatic force on the window
is equal to the integral of 9,800
times d of y, which is two point seven five minus y
time L of y, which is 2 times the square root
of point three five squared minus y squared
integrated with respect to y
from the bottom of the window to the top,
which would be from zero to point three five.
Now we could evaluate this by hand
but the main purpose of this video is to practice
setting up this integral.
So now we'll evaluate this on the graphing calculator.
But before we do this, let's go ahead and factor
out the 9,800 as well as the two.
So this would give us 19,600
times the integral of two point seven five minus y
times the square root of point three five squared
minus y squared, integrated with respect to y
from zero to zero point three five.
And now we'll go to the calculator.
So we have 19,600, math option nine
for function integration.
Now we'll enter the integrand but we'll use x instead of y.
So in parentheses we have two point seven five
minus x, closed parenthesis
times the square root of point three five squared
minus x squared, closed parenthesis
for the square root
comma x, the variable of integration comma zero
comma point three five, closed parenthesis and ENTER.
And so the hydrostatic force on the window
is approximately 4,905
point six seven six Newtons.
I hope you found this helpful.