MA 242 Unit 5 PDF

## Polar Coordinates - **Definition:** Polar coordinates are a way to represent points in a plane using a distance from the origin (r) and an angle from the positive x-axis (θ). - **Conversion:** - Cartesian to Polar: - $r = \sqrt{x^2 + y^2}$ - $\theta = arctan (\frac{y}{x})$...

## Polar Coordinates - **Definition:** Polar coordinates are a way to represent points in a plane using a distance from the origin (r) and an angle from the positive x-axis (θ). - **Conversion:** - Cartesian to Polar: - $r = \sqrt{x^2 + y^2}$ - $\theta = arctan (\frac{y}{x})$ - Polar to Cartesian: - $x = rcos(\theta)$ - $y = rsin(\theta)$ - $x^2 + y^2 = r^2 cos^2(\theta) + r^2 sin^2(\theta) = r^2$ ## Double Integrals with Polar Coordinates - **Domain:** Points in the polar plane represented by a range of radii (r) and angles (θ). - **Riemann Sum setup:** 1. For each subdivision, pick a representative point $(r_i^\ast, θ_j^\ast)$ and compute the height value: $f(r_i^\ast cos(θ_j^\ast), r_i^\ast sin(θ_j^\ast))$ 2. Determine the area of the base: - **Rectangular:** ΔA = ΔxΔy - **Polar:** ΔA = $r^\ast$ ΔrΔθ - **Derivation:** - **"Good" Area:** The area of a small wedge-shaped region in polar coordinates with radii $r_{i-1}$ and $r_i$, and angle ΔΘ, is: $$ r_i^\ast Δr ΔΘ $$ where $r_i^\ast$ is the average radius. ## Applications of Polar Coordinates ### Example 1: Mass of a Lamina - **Problem:** Calculate the mass of a lamina (thin plate) in the shape of a ring with inner radius 1 and outer radius 3, and density $σ(x, y) = x^2 + y^2$. - **Solution:** 1. Convert to polar coordinates: $x = rcos(θ)$, $y = rsin(θ)$, $dx dy = r dr dθ$, $σ(rcos(θ), rsin(θ)) = r^2$ 2. Set up the integral: $$Mass = \int_{0}^{2π} \int_{1}^{3} r^2 * r dr dθ $$ 3. Evaluate: $Mass = 40π$ ### Example 2: Double Integral Evaluation - **Problem:** Evaluate $\iint_D xy dA$, where D is the region bounded by the circle $x^2 + y^2 = 4$ and the x-axis in the first quadrant. - **Solution:** 1. Convert to polar coordinates: $x = rcos(θ)$, $y = rsin(θ)$, $dx dy = r dr dθ$ 2. Determine the integral limits: $0 ≤ r ≤ 2$, $0 ≤ θ ≤ π/2$ 3. Set up the integral: $$\iint_D xy dA = \int_{0}^{π/2} \int_{0}^{2} rcos(θ)rsin(θ) * r dr dθ $$ 4. Evaluate: $ \iint_D xy dA = \frac{15}{8} $ ### Example 3: Volume Inside a Sphere and Outside a Cylinder - **Problem:** Find the volume enclosed by the sphere $x^2 + y^2 + z^2 = 16$ and outside the cylinder $x^2 + y^2 = 4$. - **Solution:** 1. Draw a picture. 2. Determine the domain in polar coordinates: $2 ≤ r ≤ 4$, $0 ≤ θ ≤ 2π$ 3. Identify the function: $ f(x, y) = z = \sqrt{16 - x^2 - y^2} = \sqrt{16 - r^2}$ 4. Set up the integral: $Vol = 2 \int_{0}^{2π} \int_{2}^{4} \sqrt{16 - r^2} r dr dθ$ 5. Evaluate: $Vol = \frac{32π}{3} (12\sqrt{3} - 2)$ ## Polar/Cylindrical Coordinates in Triple Integrals - **Definition:** Cylindrical coordinates extend polar coordinates to three dimensions by adding the z-coordinate. - **Conversion:** - Cartesian to Cylindrical: - $r = \sqrt{x^2 + y^2}$ - $θ = arctan (\frac{y}{x})$ - $z = z$ - Cylindrical to Cartesian: - $x = rcos(θ)$ - $y = rsin(θ)$ - $z = z$ - **Triple Integral setup:** $$ \iiint_D f(x, y, z) dV = \iiint_D f(rcos(θ), rsin(θ), z) * r dr dθ dz$$ ## Spherical Coordinates in Triple Integrals - **Definition:** Spherical coordinates represent points in xyz-space using ρ, a distance from the origin; θ, an angle from the positive x-axis; and φ, an angle from the positive z-axis. - **Conversion:** - Cartesian to Spherical: - $ρ = \sqrt{x^2 + y^2 + z^2}$ - $θ = arctan(\frac{y}{x})$ - $φ = arccos(\frac{z}{ρ})$ - Spherical to Cartesian: - $x = ρsin(φ)cos(θ)$ - $y = ρsin(φ)sin(θ)$ - $z = ρcos(φ)$ ## Applications of Spherical Coordinates ### Example 1: Volume of a Spherical Wedge - **Problem:** Show that the volume of a spherical wedge (an apple slice) with radius R and angle α is $\frac{2}{3}πR^3$. - **Solution:** - **Domain:** $0 ≤ ρ ≤ R$, $0 ≤ φ ≤ π$, $α ≤ θ ≤ α + 2π$ - **Integral:** $V = \int_{α}^{α+2π} \int_{0}^{π} \int_{0}^{R} ρ^2 sin(φ) dρ dφ dθ$ - **Evaluation:** $V = \frac{2}{3}πR^3$ ### Example 2: Volume Above a Cone - **Problem:** Calculate the volume of the solid region above the cone z = - **Solution:** - **Domain:** $0 ≤ ρ ≤ \sqrt{8}$, $0 ≤ φ ≤ π/4$, $0 ≤ θ ≤ π/2$ - **Integral:** $V = \int_{0}^{π/2} \int_{0}^{π/4} \int_{0}^{√8} ρ^2 sin(φ) dρ dφ dθ$ - **Evaluation:** $V = \frac{8π}{3}(√2 - 1)$ ### Example 3: Mass of a Body - **Problem:** Find the mass of a body occupying the space between the spheres $ρ = 1$ and $ρ = 3$ with density function $σ(x, y, z) = \frac{√x^2 + y^2}{ρ}$. - **Solution:** - **Domain:** $1 ≤ ρ ≤ 3$, $0 ≤ φ ≤ π/2$, $0 ≤ θ ≤ 2π$ - **Integral:** $Mass = \int_{0}^{2π} \int_{0}^{π/2} \int_{1}^{3} \frac{ρsin(φ)}{ρ} * ρ^2 sin(φ) dρ dφ dθ$ - **Evaluation:** $Mass = 10π^2$ ### Example 4: Setting up Spherical Integrals - **Problem:** Set up a triple integral in spherical coordinates to represent the integral $\int_{-R}^{R} \int_{0}^{√{R^2 - y^2}}\int_{√{x^2 + y^2}}^{R} x dz dx dy$. - **Solution:** 1. **Determine the shape:** The integral represents a solid cone with its vertex at the origin and its base in the xy-plane. 2. **Determine the domain:** - ρ: $0 ≤ ρ ≤ R/cos(φ)$ (cone equation) - θ: $-\frac{π}{2} ≤ θ ≤ \frac{π}{2}$ - φ: $0 ≤ φ ≤ \frac{π}{4}$ (cone equation) 3. **Convert x:** $x = ρsin(φ)cos(θ)$ 4. **Set up the integral:** $$ \int_{0}^{π/4} \int_{-π/2}^{π/2} \int_{0}^{R/cos(φ)} ρsin(φ)cos(θ) * ρ^2sin(φ) dρ dθ dφ$$ The provided text presents a comprehensive overview of polar and spherical coordinates, including: * Their definitions and conversions to and from Cartesian coordinates * Setting up double and triple integrals in polar and spherical coordinates * Detailed examples with steps and explanations * Applications including: calculating mass, volume, and evaluating integrals. This information would be helpful for students or anyone looking to understand and apply these coordinate systems in various mathematical contexts.