Day 27

The solution is $\ds\iint_Rdm =\iint_R\delta dydx = \int_0^3\int_0^{9-x^2}(xy)dydx$.

Here's some easy to modify Mathematica code that will compute the integral for you, as well as draw the region to help you know that what you gave for the bounds is correct.

OuterBounds = {x, 0, 3};
InnerBounds = {y, 0, 9 - x^2};

Integrand = x y;
Integrate[Integrand, OuterBounds, InnerBounds]

CoordinateSystem = {x, y};
ParametricPlot[CoordinateSystem, OuterBounds, InnerBounds, Mesh -> {10, 0}]
ParametricPlot[Evaluate[CoordinateSystem, OuterBounds, InnerBounds], Mesh -> {10, 0}]

We have $\ds\bar x = \frac{\iint_Rxdm}{\iint_Rdm} =\frac{\iint_Rxdydx}{\iint_Rdydx} = \frac{\int_0^3\int_0^{9-x^2}x(xy)dydx}{\int_0^3\int_0^{9-x^2}xydydx}$.

The only difference for $\bar y$ is we change the $x$ to a $y$, so that we are finding the average $y$ value, instead of the average $x$-value. This gives $\ds\bar y = \frac{\iint_Rydm}{\iint_Rdm} =\frac{\iint_Rydydx}{\iint_Rdydx} = \frac{\int_0^3\int_0^{9-x^2}y(xy)dydx}{\int_0^3\int_0^{9-x^2}xydydx}$.

OuterBounds = {x, 0, 3};
InnerBounds = {y, 0, 9 - x^2};

density = x y;
Integrate[x*density, OuterBounds, InnerBounds]/Integrate[density, OuterBounds, InnerBounds]
Integrate[y*density, OuterBounds, InnerBounds]/Integrate[density, OuterBounds, InnerBounds]

CoordinateSystem = {x, y};
ParametricPlot[CoordinateSystem, OuterBounds, InnerBounds, Mesh -> {10, 0}]
ParametricPlot[Evaluate[CoordinateSystem, OuterBounds, InnerBounds], Mesh -> {10, 0}]

The three integral notations all add up area. The first notation assumes you have cut the region up into tiny rectangles where both the width and height have been reduced. The second slices a region up into vertical slabs with a tiny width $dx$. The later assumes the slabs are horizontal, with a tiny height $dy$. They all compute the same number, just in a different way. Fubini's theorem provides precise details under which these three integrals are guaranteed to be equal.

We can use the Mesh command in Mathematica to quickly visualize this.

xBounds = {x, 0, 3};
yBounds = {y, 0, 9 - x^2};

CoordinateSystem = {x, y};
ParametricPlot[Evaluate[CoordinateSystem, xBounds, yBounds], Mesh -> {20, 20}]
ParametricPlot[Evaluate[CoordinateSystem, xBounds, yBounds], Mesh -> {20, 0}]
ParametricPlot[Evaluate[CoordinateSystem, xBounds, yBounds], Mesh -> {0, 20}]

The area is $\ds\iint_RdA =\iint_Rr drd\theta = \int_{\pi/2}^{\pi}\int_0^{5}rdrd\theta$.

Here's some code that will plot any polar region. The key is to use the change-of-coordinates $x = r\cos\theta$ and $y=r\sin\theta$. This code can be adapted to work with any change of coordinates, which we'll focus on in a couple weeks.

OuterBounds = {theta, Pi/2, Pi};
InnerBounds = {r, 0, 5};

Integrand = r;
Integrate[Integrand, OuterBounds, InnerBounds]

CoordinateSystem = {r Cos[theta], r Sin[theta]};
ParametricPlot[CoordinateSystem, OuterBounds, InnerBounds, Mesh -> {10, 0}]
ParametricPlot[Evaluate[CoordinateSystem, OuterBounds, InnerBounds], Mesh -> {10, 0}]

The centroid $(\bar x, \bar y)$ is the point that gives the average $(x,y)$ coordinates of the points in the region. It's the geometric center, and variable densities are ignored. These are just average $x$ and $y$ values, so we use the average value formula $\bar f = \frac{\iint_R f dA}{\iint_R dA}$, with $f$ equal to $x$ and $y$ respectively, to obtain

• $\ds\bar x = \frac{\iint_RxdA}{\iint_RdA} =\frac{\iint_R(r\cos\theta)rdrd\theta}{\iint_Rrdrd\theta} = \frac{\int_{\pi/2}^{\pi}\int_0^{5}rdrd\theta}{\int_{\pi/2}^{\pi}\int_0^{5}(r\cos\theta)rdrd\theta}$ and
• $\ds\bar y = \frac{\iint_RxdA}{\iint_RdA} =\frac{\iint_R(r\sin\theta)rdrd\theta}{\iint_Rrdrd\theta} = \frac{\int_{\pi/2}^{\pi}\int_0^{5}rdrd\theta}{\int_{\pi/2}^{\pi}\int_0^{5}(r\sin\theta)rdrd\theta}$.

With Mathematica, we can quickly compute these.

OuterBounds = {theta, Pi/2, Pi};
InnerBounds = {r, 0, 5};

Integrate[(r Cos[theta]) r, OuterBounds, InnerBounds]/Integrate[r, OuterBounds, InnerBounds]
Integrate[(r Sin[theta]) r, OuterBounds, InnerBounds]/Integrate[r, OuterBounds, InnerBounds]

CoordinateSystem = {r Cos[theta], r Sin[theta]};
ParametricPlot[CoordinateSystem, OuterBounds, InnerBounds, Mesh -> {10, 0}]
ParametricPlot[Evaluate[CoordinateSystem, OuterBounds, InnerBounds], Mesh -> {10, 0}]