Session 5
Lectures 18 - 22
- 14.8 Lagrange Multipliers (revisited)
- 15.1 Double Integrals over Rectangles
- 15.2 Double Integrals over General Regions
- 15.3 Double Integrals in Polar Coordinates
14.8 Lagrange Multipliers (revisited)
To find the extreme values of \(f(x,y,z)\) subject to the constraint \(g(x,y,z) = k\) (assuming they exist and \(\nabla g \neq \mathbf{0}\) on the constraint):
- Find all \(x\), \(y\), \(z\), and \(\lambda\) such that \[
\nabla f(x,y,z) = \lambda \nabla g(x,y,z) \qquad g(x,y,z) = k
\]
- Evaluate \(f\) at all the points found in step 1. The largest is the maximum, the smallest is the minimum
- Geometrically, at an extremum the level curve/surface of \(f\) is tangent to the constraint, so \(\nabla f \parallel \nabla g\)
14.8 Lagrange Multipliers (revisited)
cont.
Two Constraints
- To find extreme values of \(f(x,y,z)\) subject to \(g(x,y,z) = k\) and \(h(x,y,z) = c\), solve \[
\nabla f = \lambda \nabla g + \mu \nabla h \qquad g(x,y,z) = k \qquad h(x,y,z) = c
\]
Combining with 14.7
- To find absolute extrema on a closed region, use Lagrange multipliers to handle the boundary, and compare with the critical points in the interior
Exercises
14.8 Lagrange Multipliers
- Find the maximum and minimum values of the function \(f(x,y) = x + y\) on the curve \(x^2 + y^2 - xy = 4\)
- Consider a hypothetical planet whose surface can be approximated by the sphere \(x^2 + y^2 + z^2 = 1\). Suppose the planet’s temperature is strange and given by \(T(x,y,z) = 2x + 2y + z\). Find the maximum and minimum temperatures on the planet’s surface
15.1 Double Integrals over Rectangles
Let \(R = [a,b] \times [c,d] = \{(x,y) \mid a \le x \le b,\ c \le y \le d\}\). The double integral of \(f\) over \(R\) is \[
\iint_R f(x,y)\,dA = \lim_{m,n \to \infty} \sum_{i=1}^{m} \sum_{j=1}^{n} f(x_{ij}^*, y_{ij}^*)\,\Delta A
\]
- If \(f(x,y) \ge 0\), then \(\iint_R f(x,y)\,dA\) is the volume of the solid above \(R\) and below the surface \(z = f(x,y)\)
- The average value of \(f\) over \(R\) is \(f_{\text{ave}} = \frac{1}{A(R)} \iint_R f(x,y)\,dA\)
15.1 Double Integrals over Rectangles
cont.
Fubini’s Theorem
If \(f\) is continuous on \(R = [a,b] \times [c,d]\), then \[
\iint_R f(x,y)\,dA = \int_a^b \int_c^d f(x,y)\,dy\,dx = \int_c^d \int_a^b f(x,y)\,dx\,dy
\]
- Work from the inside out: integrate the inner variable while treating the other as a constant
- The order does not change the answer, but it can make the integral much easier (or harder)
- If \(f(x,y) = g(x)h(y)\), then \(\iint_R g(x)h(y)\,dA = \left(\int_a^b g(x)\,dx\right)\left(\int_c^d h(y)\,dy\right)\)
Exercises
15.1 Double Integrals over Rectangles
- Evaluate \(\iint_R x y^2\,dA\) where \(R = [0,2] \times [1,3]\)
- Evaluate \(\iint_R y e^{xy}\,dA\) where \(R = [0,2] \times [0,1]\). Which order of integration is easier?
15.2 Double Integrals over General Regions
Type I: \(D = \{(x,y) \mid a \le x \le b,\ g_1(x) \le y \le g_2(x)\}\) \[
\iint_D f(x,y)\,dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx
\]
Type II: \(D = \{(x,y) \mid c \le y \le d,\ h_1(y) \le x \le h_2(y)\}\) \[
\iint_D f(x,y)\,dA = \int_c^d \int_{h_1(y)}^{h_2(y)} f(x,y)\,dx\,dy
\]
- Always sketch the region first. The outer limits must be constants, the inner limits can depend on the outer variable
15.2 Double Integrals over General Regions
cont.
Properties
- \(\iint_D [f(x,y) + g(x,y)]\,dA = \iint_D f(x,y)\,dA + \iint_D g(x,y)\,dA\) and \(\iint_D c f(x,y)\,dA = c\iint_D f(x,y)\,dA\)
- If \(D = D_1 \cup D_2\) (not overlapping except on boundaries), then \(\iint_D f\,dA = \iint_{D_1} f\,dA + \iint_{D_2} f\,dA\)
- \(\iint_D 1\,dA = A(D)\), the area of \(D\)
- If \(m \le f(x,y) \le M\) on \(D\), then \(m\,A(D) \le \iint_D f(x,y)\,dA \le M\,A(D)\)
Exercises
15.2 Double Integrals over General Regions
- Let \(D\) be the triangular region bounded by \(x = 0\), \(y = 0\), and \(x + y = 1\)
- What are the three vertices of \(D\)?
- Evaluate \(\iint_D xy\,dA\)
- Compute \(\displaystyle\int_0^{1/2} \int_{y\sqrt{3}}^{\sqrt{1-y^2}} 30xy^2\,dx\,dy\)
15.3 Double Integrals in Polar Coordinates
Recall \(x = r\cos\theta\), \(y = r\sin\theta\), and \(r^2 = x^2 + y^2\)
If \(f\) is continuous on the polar rectangle \(R = \{(r,\theta) \mid 0 \le a \le r \le b,\ \alpha \le \theta \le \beta\}\) where \(0 \le \beta - \alpha \le 2\pi\), then \[
\iint_R f(x,y)\,dA = \int_\alpha^\beta \int_a^b f(r\cos\theta, r\sin\theta)\,r\,dr\,d\theta
\]
- Don’t forget the extra factor of \(r\): \(dA = r\,dr\,d\theta\)
- Polar is usually the right choice when the region is a disk, annulus, or sector, or when \(f\) involves \(x^2 + y^2\)
15.3 Double Integrals in Polar Coordinates
cont.
General Polar Regions
If \(D = \{(r,\theta) \mid \alpha \le \theta \le \beta,\ h_1(\theta) \le r \le h_2(\theta)\}\), then \[
\iint_D f(x,y)\,dA = \int_\alpha^\beta \int_{h_1(\theta)}^{h_2(\theta)} f(r\cos\theta, r\sin\theta)\,r\,dr\,d\theta
\]
Exercises
15.3 Double Integrals in Polar Coordinates
- Consider \(f(x,y) = x - y\) on the region \(D\) defined by \(x + y \ge 0\) and \(x^2 + y^2 \le 4\)
- Sketch the region \(D\)
- Calculate the integral of \(f\) over \(D\)
Exercises
15.3 Double Integrals in Polar Coordinates (cont.)
- Let \(R\) be the region inside the circle \(x^2 + (y-1)^2 = 1\) but outside the circle \(x^2 + y^2 = 1\)
- Find the bounds of \(R\) in polar coordinates
- A lamina in the shape of \(R\) has mass density \(f(x,y) = \dfrac{1}{\sqrt{x^2+y^2}} - \dfrac{x}{x^2+y^2}\). Find the total mass \(M\) of the lamina
Fin
This concludes session 5.