Polar Graphing

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Answer Key & Explanations

Polar Graphing · Grade 12 · Worksheet 1

1. ∫(x³ - 3x² + 2x) dx from 0 to 2 = ? Answer: 0 Solution: Find the antiderivative of x³ - 3x² + 2x Antiderivative = (1/4)x⁴ - x³ + x² Evaluate at the upper limit (x = 2) F(2) = (1/4)(2)⁴ - (2)³ + (2)² = (1/4)(16) - 8 + 4 = 4 - 8 + 4 = 0 Evaluate at the lower limit (x = 0) F(0) = (1/4)(0)⁴ - (0)³ + (0)² = 0 - 0 + 0 = 0 ∫(x³ - 3x² + 2x) dx from 0 to 2 =…
   Full step-by-step solution

   Step 1: Find the antiderivative of x³ - 3x² + 2x Antiderivative = (1/4)x⁴ - x³ + x² Step 2: Evaluate at the upper limit (x = 2) F(2) = (1/4)(2)⁴ - (2)³ + (2)² = (1/4)(16) - 8 + 4 = 4 - 8 + 4 = 0 Step 3: Evaluate at the lower limit (x = 0) F(0) = (1/4)(0)⁴ - (0)³ + (0)² = 0 - 0 + 0 = 0 Step 4: Apply the Fundamental Theorem of Calculus ∫(x³ - 3x² + 2x) dx from 0 to 2 = F(2) - F(0) = 0 - 0 = 0 The answer is 0.

2. Graph r = 5 + 5cos(θ) and identify the shape. Answer: cardioid Solution: Identify the general form. The equation r = 5 + 5cos(θ) is of the form r = a + bcos(θ) with a = 5 and b = 5. Determine the ratio a/b = 5/5 = 1.
   Full step-by-step solution

   Step 1: Identify the general form. The equation r = 5 + 5cos(θ) is of the form r = a + bcos(θ) with a = 5 and b = 5. Step 2: Determine the ratio a/b = 5/5 = 1. Step 3: When a/b = 1, the graph is a cardioid (a special type of limaçon with a cusp). Step 4: The graph is symmetric about the polar axis (since cosine is even). Step 5: Key points: At θ = 0, r = 5 + 5(1) = 10. At θ = π/2, r = 5 + 5(0) = 5. At θ = π, r = 5 + 5(-1) = 0 (the cusp). At θ = 3π/2, r = 5 + 5(0) = 5. Step 6: The shape is a cardioid. The answer is cardioid.

3. ∫(x²e^x) dx from 0 to 1 = ? Answer: e-2 Solution: Use integration by parts. Let u = x² and dv = e^x dx Then du = 2x dx and v = e^x Apply integration by parts: ∫x²e^x dx = x²e^x - ∫2xe^x dx Apply integration by parts again to ∫2xe^x dx.
   Full step-by-step solution

   Step 1: Use integration by parts. Let u = x² and dv = e^x dx Step 2: Then du = 2x dx and v = e^x Step 3: Apply integration by parts: ∫x²e^x dx = x²e^x - ∫2xe^x dx Step 4: Apply integration by parts again to ∫2xe^x dx. Let u = 2x and dv = e^x dx Step 5: Then du = 2 dx and v = e^x Step 6: ∫2xe^x dx = 2xe^x - ∫2e^x dx = 2xe^x - 2e^x Step 7: Substitute back: ∫x²e^x dx = x²e^x - (2xe^x - 2e^x) = x²e^x - 2xe^x + 2e^x Step 8: Evaluate from 0 to 1: [1²e¹ - 2(1)e¹ + 2e¹] - [0²e⁰ - 2(0)e⁰ + 2e⁰] Step 9: = [e - 2e + 2e] - [0 - 0 + 2(1)] Step 10: = [e] - [2] = e - 2 The answer is e-2.

4. Kaia is analyzing a polar graph described by the equation r = 9 sin(3θ). Identify the type of polar curve this equation produces, determine the total number of petals, and find the exact polar coordinates (r, θ) of the tip of one petal located in the first quadrant where θ is the smallest positive angle. Answer: (9, π/6) Solution: Identify the curve type. The equation r = 9 sin(3θ) is of the form r = a sin(nθ) with a = 9 and n = 3. Since n = 3 is odd, the rose curve has n = 3 petals.
   Full step-by-step solution

   Step 1: Identify the curve type. The equation r = 9 sin(3θ) is of the form r = a sin(nθ) with a = 9 and n = 3. Since n = 3 is odd, the rose curve has n = 3 petals. Step 2: Petal tips occur when sin(3θ) = ±1, giving the maximum distance from the origin, r = 9. Step 3: Solve sin(3θ) = 1 for the smallest positive θ. The general solution is 3θ = π/2 + 2πk, so θ = π/6 + (2π/3)k, where k is an integer. Step 4: Find the smallest positive θ in the first quadrant (0 < θ < π/2). For k = 0, θ = π/6. This is in the first quadrant. For k = 1, θ = π/6 + 2π/3 = 5π/6, which is in the second quadrant. So the smallest positive θ is π/6. Step 5: Substitute θ = π/6 into the equation: r = 9 sin(3 * π/6) = 9 sin(π/2) = 9(1) = 9. Step 6: The polar coordinates of this petal tip are (9, π/6). The answer is (9, π/6).

5. Graph r = 2 + 2sin(θ) and identify the shape. Determine the maximum value of r. Answer: cardioid, 4 Solution: Recognize the equation r = 2 + 2sin(θ) is of the form r = a + b sin(θ) with a = 2 and b = 2. Since a = b, this is a cardioid. The cardioid is symmetric about the vertical axis (θ = π/2) because of the sine function.
   Full step-by-step solution

   Step 1: Recognize the equation r = 2 + 2sin(θ) is of the form r = a + b sin(θ) with a = 2 and b = 2. Since a = b, this is a cardioid. Step 2: The cardioid is symmetric about the vertical axis (θ = π/2) because of the sine function. Step 3: To find the maximum value of r, note that sin(θ) ranges from -1 to 1. The maximum occurs when sin(θ) = 1. Step 4: Substitute sin(θ) = 1 into the equation: r = 2 + 2(1) = 4. Step 5: The graph is a cardioid with maximum r = 4 at θ = π/2. The answer is cardioid, 4.

6. Graph r = 5 + 5cosθ and identify the shape. Answer: cardioid Solution: Recognize the equation r = 5 + 5cosθ is in the form r = a + bcosθ, where a = 5 and b = 5. Since a = b, the graph is a cardioid (a special limaçon with a cusp). The graph is symmetric about the polar axis (cosθ symmetry).
   Full step-by-step solution

   Step 1: Recognize the equation r = 5 + 5cosθ is in the form r = a + bcosθ, where a = 5 and b = 5. Step 2: Since a = b, the graph is a cardioid (a special limaçon with a cusp). Step 3: The graph is symmetric about the polar axis (cosθ symmetry). Step 4: Key points: when θ = 0, r = 5 + 5(1) = 10; when θ = π/2, r = 5 + 5(0) = 5; when θ = π, r = 5 + 5(-1) = 0 (cusp at the pole). Step 5: The shape is a heart-shaped curve (cardioid) opening to the right. The answer is cardioid.

7. A polar graph is described by the equation r = 2 + 4sin(θ). This limaçon has an inner loop. Determine the exact area enclosed by the inner loop of this polar curve. Answer: 4π - 6√3 Solution: Find where r = 0 to determine the limits for the inner loop. 2 + 4sin(θ) = 0 4sin(θ) = -2 sin(θ) = -1/2 θ = 7π/6 and 11π/6 Set up the area integral for the inner loop.
   Full step-by-step solution

   Step 1: Find where r = 0 to determine the limits for the inner loop. 2 + 4sin(θ) = 0 4sin(θ) = -2 sin(θ) = -1/2 θ = 7π/6 and 11π/6 Step 2: Set up the area integral for the inner loop. A = 1/2 ∫[7π/6 to 11π/6] (2 + 4sin(θ))^2 dθ Step 3: Expand the integrand. (2 + 4sin(θ))^2 = 4 + 16sin(θ) + 16sin^2(θ) Step 4: Use the identity sin^2(θ) = (1 - cos(2θ))/2 16sin^2(θ) = 16(1 - cos(2θ))/2 = 8 - 8cos(2θ) Step 5: Rewrite the integrand. 4 + 16sin(θ) + 8 - 8cos(2θ) = 12 + 16sin(θ) - 8cos(2θ) Step 6: Integrate term by term from 7π/6 to 11π/6. A = 1/2 ∫[7π/6 to 11π/6] (12 + 16sin(θ) - 8cos(2θ)) dθ Step 7: Compute the antiderivative. ∫12 dθ = 12θ ∫16sin(θ) dθ = -16cos(θ) ∫-8cos(2θ) dθ = -4sin(2θ) Step 8: Evaluate from 7π/6 to 11π/6. At θ = 11π/6: 12(11π/6) - 16cos(11π/6) - 4sin(11π/3) = 22π - 16(√3/2) - 4(-√3/2) = 22π - 8√3 + 2√3 = 22π - 6√3 At θ = 7π/6: 12(7π/6) - 16cos(7π/6) - 4sin(7π/3) = 14π - 16(-√3/2) - 4(√3/2) = 14π + 8√3 - 2√3 = 14π + 6√3 Step 9: Subtract and multiply by 1/2. A = 1/2[(22π - 6√3) - (14π + 6√3)] = 1/2[8π - 12√3] = 4π - 6√3 The exact area enclosed by the inner loop is 4π - 6√3.

8. Moana is tracking the orbit of a satellite using polar coordinates. The satellite's trajectory is modeled by the polar equation r = 12 + 6cos(θ). If the satellite reaches its closest point to Earth when θ = π, what is that minimum distance r from Earth's center? Answer: 6 Solution: The polar equation is r = 12 + 6cos(θ). At the closest point, θ = π, so cos(π) = -1. Substitute: r = 12 + 6 * (-1) = 12 - 6.
   Full step-by-step solution

   Step 1: The polar equation is r = 12 + 6cos(θ). Step 2: At the closest point, θ = π, so cos(π) = -1. Step 3: Substitute: r = 12 + 6 * (-1) = 12 - 6. Step 4: Calculate: 12 - 6 = 6. The minimum distance from Earth's center is 6 units.

9. ∫(e^(2x) + 3x²) dx from 0 to 1 = ? Answer: (e² - 1)/2 + 1 Solution: Break the integral into two parts: ∫e^(2x) dx from 0 to 1 + ∫3x² dx from 0 to 1 For ∫e^(2x) dx, use substitution: u = 2x, du = 2 dx → (1/2)∫e^u du = (1/2)e^u = (1/2)e^(2x) Evaluate from 0 to 1: (1/2)e^(2) - (1/2)e^0 = (1/2)(e² - 1) For ∫3x² dx = 3∫x² dx = 3(x³/3) = x³ Evaluate from 0 to 1: 1³ -…
   Full step-by-step solution

   Step 1: Break the integral into two parts: ∫e^(2x) dx from 0 to 1 + ∫3x² dx from 0 to 1 Step 2: For ∫e^(2x) dx, use substitution: u = 2x, du = 2 dx → (1/2)∫e^u du = (1/2)e^u = (1/2)e^(2x) Step 3: Evaluate from 0 to 1: (1/2)e^(2) - (1/2)e^0 = (1/2)(e² - 1) Step 4: For ∫3x² dx = 3∫x² dx = 3(x³/3) = x³ Step 5: Evaluate from 0 to 1: 1³ - 0³ = 1 Step 6: Combine results: (1/2)(e² - 1) + 1 = (e² - 1)/2 + 1 The answer is (e² - 1)/2 + 1.