Session 6
Lectures 23 - 25
- 15.4 Applications of Double Integrals
- 15.5 Surface Area
- 15.6 Triple Integrals
- 15.7 Triple Integrals in Cylindrical Coordinates
15.4 Applications of Double Integrals
Suppose a lamina occupies a region \(D\) in the \(xy\)-plane and has density \(\rho(x,y)\) (mass per unit area)
- The total mass is \[
m = \iint_D \rho(x,y)\,dA
\]
15.4 Applications of Double Integrals
cont.
Moments and Center of Mass
- The moment about the \(x\)-axis and the moment about the \(y\)-axis are \[
M_x = \iint_D y\,\rho(x,y)\,dA \qquad M_y = \iint_D x\,\rho(x,y)\,dA
\]
- The center of mass \((\bar{x}, \bar{y})\) is \[
\bar{x} = \frac{M_y}{m} = \frac{1}{m}\iint_D x\,\rho(x,y)\,dA \qquad \bar{y} = \frac{M_x}{m} = \frac{1}{m}\iint_D y\,\rho(x,y)\,dA
\]
Exercises
15.4 Applications of Double Integrals
- Consider the half-disc \(0 \le x^2 + y^2 \le 1\) with \(y \ge 0\). Assume that the density is \(\rho(x,y) = 1\)
- Find \(\bar{y}\), the \(y\)-coordinate of the center of mass of the half-disc
- Find the \(y\)-coordinate of the center of mass of the triangular region with vertices \((-\tfrac{1}{2}, 0)\), \((\tfrac{1}{2}, 0)\), and \((0, \tfrac{1}{2})\), with the same density
15.5 Surface Area
The area of the surface \(z = f(x,y)\) over a region \(D\), where \(f_x\) and \(f_y\) are continuous, is \[
A(S) = \iint_D \sqrt{1 + [f_x(x,y)]^2 + [f_y(x,y)]^2}\,dA
\]
- Equivalently, \(A(S) = \displaystyle\iint_D \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2}\,dA\)
- Compare with arc length \(L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx\)
Exercises
15.5 Surface Area
- Show that the area of the part of the plane \(z = ax+by+c\) that projects onto a region \(D\) in the \(xy\)-plane with area \(A(D)\) is \(A(D)\sqrt{a^2+b^2+1}\).
15.6 Triple Integrals
Fubini’s Theorem for a box \(B = [a,b] \times [c,d] \times [r,s]\): \[
\iiint_B f(x,y,z)\,dV = \int_r^s \int_c^d \int_a^b f(x,y,z)\,dx\,dy\,dz
\]
- Any of the six orders of integration gives the same answer
- \(\iiint_E 1\,dV = V(E)\), the volume of \(E\)
- Mass and center of mass work just like 15.4: \(m = \iiint_E \rho\,dV\), \(\bar{x} = \frac{1}{m}\iiint_E x\rho\,dV\), etc.
Exercises
15.6 Triple Integrals
- Evaluate \(\iiint_B xyz\,dV\) for the following choices of \(B\)
- \(B\) is the cube \(0 \le x \le 1\), \(0 \le y \le 1\), \(0 \le z \le 1\)
- \(B\) is the region \(0 \le x \le y \le z \le 1\)
15.7 Triple Integrals in Cylindrical Coordinates
Cylindrical coordinates are polar coordinates in the \(xy\)-plane with \(z\) left alone: \[
x = r\cos\theta \qquad y = r\sin\theta \qquad z = z \qquad r^2 = x^2 + y^2
\]
If \(E = \{(x,y,z) \mid (x,y) \in D,\ u_1(x,y) \le z \le u_2(x,y)\}\) and \(D\) is given in polar by \(\alpha \le \theta \le \beta\), \(h_1(\theta) \le r \le h_2(\theta)\), then \[
\iiint_E f\,dV = \int_\alpha^\beta \int_{h_1(\theta)}^{h_2(\theta)} \int_{u_1(r\cos\theta,\, r\sin\theta)}^{u_2(r\cos\theta,\, r\sin\theta)} f(r\cos\theta, r\sin\theta, z)\,r\,dz\,dr\,d\theta
\]
- Again, don’t forget the \(r\): \(dV = r\,dz\,dr\,d\theta\)
- Good choice when \(E\) has symmetry about the \(z\)-axis (cylinders, cones, paraboloids) or \(f\) involves \(x^2 + y^2\)
Exercises
15.7 Triple Integrals in Cylindrical Coordinates
- Find the volume of the solid bounded by the paraboloids \(z = x^2 + y^2\) and \(z = 2 - x^2 - y^2\) using cylindrical coordinates
Fin
This concludes session 6.