Task 30.1

When we use double integrals to find centroids, the formulas for the centroid are similar for both $\bar x$ and $\bar y$. In other courses, you may see the formulas that appear in this task, because the ideas are presented without requiring knowledge of double integrals. Integrating the inside integral from the double integral formula gives the single variable formulas that you'll find in other courses.

Let $R$ be the region in the plane with $a\leq x\leq b$ and $g(x)\leq y\leq f(x)$.

  1. Set up an iterated integral to compute the area of $R$. Then compute the inside integral. You should obtain a familiar formula from first-semester calculus.
  2. Set up an iterated integral formula to compute $\bar x$ for the centroid. By computing the inside integrals, show that $$\ds\bar x = \frac{\int_a^b x (f-g)dx}{\int_a^b (f-g)dx}.$$
  3. If the density depends only on $x$, so $\delta = \delta (x)$, set up an iterated integral formula to compute $\bar y$ for the center of mass. Compute the inside integral and show that $$\ds\bar y = \frac{\ds\int_a^b \frac{1}{2}(f^2-g^2)\delta(x)dx}{\ds\int_a^b (f-g)\delta(x)dx} = \frac{\ds\int_a^b \overbrace{\frac{(f+g)}{2}}^{\tilde y}\overbrace{\delta(x)\underbrace{(f-g)dx}_{dA}}^{dm}}{\text{mass}}.$$

In class, fell free to ask and we'll analyze the integral formula above and show how you can set this up as a single integral using geometric reasoning. We'll discuss the quantities $\tilde y$, $dm$, and $dA$, as appropriate.

Task 30.2

We've been working with rods, wires, thin plates, and solid domains. For example, we could work with a circular wire, or a circular disc, or a ball. How do the centroid formulas change in each setting? This task has us examine these three setting, set up the corresponding integrals, use software to solve them, and then compare the locations of the centroids.

Consider the curve $C$ that is the upper half of the circle $x^2+y^2 = 49$, the region $R$ that lies above $y=0$ and inside the circle $x^2+y^2=49$, and the solid domain $D$ that lies inside the sphere $x^2+y^2+z^2=49$ and satisfies $y\geq 0$. Because of symmetry, for each region it is clear that $\bar x=\bar z=0$.

  1. Set up an integral formula to compute $\bar y$ for the curve $C$. [Hint: You'll need a parametrization.]
  2. Set up an integral formula to compute $\bar y$ for the region $R$. [You can use Cartesian coordinates or polar coordinates.]
  3. Set up an integral formula to compute $\bar y$ for the domain $D$.
  4. Use software to compute all three integral formulas above, obtaining an exact value for the answer (not a numerical approximation).
  5. For each object, state a general formulas for $\bar y$ if the radius is $a$ (not $7$).

Task 30.3

Pick a few regions from Wikipedia's List of Centroids and then set up and compute (with software) iterated integral formulas to find the centroids from that list. Try doing some that are 2D, and some that are 3D.

Task 30.4

Pick some problems related to the topics we are discussing from the Text Book Practice page.