Today we looked at two applications of integrals. First we looked at finding areas between curves. We did these two examples:
Find the area between the parabola \(y = 2-x^2\) and the line \(y = x\).
Find the area between \(y = \sin x\) and \(y = \cos x\).
Then we discussed how to use integrals to find volumes. We started with the problem of finding the volume of a pyramid, and came up with the central idea: that the volume of a solid is the integral of the areas of its cross sections.
\[\text{Volume} = \int \text{Area} \, dx.\]
If the cross sections are disks, this formula becomes \(V = \displaystyle \int \pi r^2 \, dx\).
Here are the examples we did in class:
Find the volume of a square pyramid if the base is 100 meters by 100 meters, and the height is 50 meters.
Find the volume of the solid formed by revolving the region between \(y = \sqrt{x}\) and the x-axis from \(x=0\) to \(x=4\) around the \(x\)-axis.
Use the disk method to find the volume of a cone that is \(h\) units tall and has a base radius of \(R\) units.
Find the volume of a sphere with radius \(R\) by revolving \(y = \sqrt{R^2 - x^2}\) around the x-axis. Here is a pretty good video explanation.
The last example we did was a hollow shape. To get the volume, you use the washers method since the cross sections look like flat washers. The formula for the washers method is:
\[ V= \int \pi R^2 - \pi r^2 \, dx\]
Today we talked about how to find the length of a curve using integrals. There are two approaches. For a function with an explicit formula \(y = f(x)\), use:
\[\text{Length} = \int_a^b \sqrt{1+(y')^2} \, dx.\]
For functions with a parametric formula where both the x and y coordinates are functions of a third parameter \(t\), use:
\[\text{Length} = \int_a^b \sqrt{(dx/dt)^2+(dy/dt)^2} \, dt.\]
This formula has a nice interpretation if \(t\) represents time. Then \((dx/dt)\) is the horizontal velocity, \((dy/dt)\) is the vertical velocity, and \(\displaystyle \sqrt{(dx/dt)^2+(dy/dt)^2}\) is the speed of the object. Then distance traveled is the integral of speed with respect to time.
Here are some video examples:
How to find arc length of \(y = x^{3/2}\) from \(x=0\) to \(x = \tfrac{32}{9}\)
Arc length of parametric curve \(x=\cos(t)\), \(y = \sin(t)\) from \(t=0\) to \(t=\frac{\pi}{2}\).
In addition to the two examples in the videos, we did the following other examples in class:
Write down an integral that represents the length of the parabola \(y = x^2\) from \(x = -2\) to \(x=2\).
Calculate the arc length of the parametric curve \(x(t) = \cos t\) and \(y(t) = t + \sin t\) from \(t = 0\) to \(t = \pi\). Hint: This one needed the half-angle formula \(\displaystyle \cos^2 \left(\frac{t}{2} \right) = \frac{1+\cos t}{2}\).
Write down an integral that represents the perimeter of the ellipse with parametric equations \(x(t) = 2 \cos t\), \(y(t) = \sin t\).
Use a Riemann sum with \(N=1000\) rectangles to estimate the perimeter of the ellipse in the previous problem.
The parametric curve \(x(t) = \ln(\sec t + \tan t)\), \(y(t) = \sec t\) is a catenary which is the shape of a hanging cable connecting two points. Find the length of this catenary from \(t = -\frac{\pi}{4}\) to \(t = \frac{\pi}{4}\).