Polar Graphing

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Polar Graphing · Grade 12 · Worksheet 2

1. ∫(x³ - 2x² + 5) dx from -1 to 2 = ? Answer: 12.75 Solution: Find the antiderivative of x³ - 2x² + 5 Antiderivative = (1/4)x⁴ - (2/3)x³ + 5x Evaluate at upper bound x = 2 F(2) = (1/4)(2)⁴ - (2/3)(2)³ + 5(2) F(2) = (1/4)(16) - (2/3)(8) + 10 F(2) = 4 - 16/3 + 10 F(2) = 14 - 16/3 F(2) = 42/3 - 16/3 = 26/3 Evaluate at lower bound x = -1 F(-1) = (1/4)(-1)⁴ -…
   Full step-by-step solution

   Step 1: Find the antiderivative of x³ - 2x² + 5 Antiderivative = (1/4)x⁴ - (2/3)x³ + 5x Step 2: Evaluate at upper bound x = 2 F(2) = (1/4)(2)⁴ - (2/3)(2)³ + 5(2) F(2) = (1/4)(16) - (2/3)(8) + 10 F(2) = 4 - 16/3 + 10 F(2) = 14 - 16/3 F(2) = 42/3 - 16/3 = 26/3 Step 3: Evaluate at lower bound x = -1 F(-1) = (1/4)(-1)⁴ - (2/3)(-1)³ + 5(-1) F(-1) = (1/4)(1) - (2/3)(-1) - 5 F(-1) = 1/4 + 2/3 - 5 F(-1) = 3/12 + 8/12 - 60/12 F(-1) = (3 + 8 - 60)/12 = -49/12 Step 4: Subtract F(-1) from F(2) 26/3 - (-49/12) = 26/3 + 49/12 Convert to common denominator: 104/12 + 49/12 = 153/12 = 51/4 = 12.75 The answer is 12.75.

2. Graph r = 7 + 7sin(θ) and identify the shape. Answer: Cardioid Solution: Recognize the equation r = 7 + 7sin(θ) is in the form r = a + b sin(θ) with a = 7 and b = 7. Since a = b, the graph is a cardioid.
   Full step-by-step solution

   Step 1: Recognize the equation r = 7 + 7sin(θ) is in the form r = a + b sin(θ) with a = 7 and b = 7. Since a = b, the graph is a cardioid. Step 2: The cardioid is symmetric about the vertical line (θ = π/2) because it uses the sine function. The maximum value of r occurs when sin(θ) = 1, giving r = 7 + 7(1) = 14. The minimum value occurs when sin(θ) = -1, giving r = 7 + 7(-1) = 0 (a cusp at the origin). Step 3: Key points: - At θ = 0: r = 7 + 7(0) = 7 - At θ = π/2: r = 7 + 7(1) = 14 - At θ = π: r = 7 + 7(0) = 7 - At θ = 3π/2: r = 7 + 7(-1) = 0 Step 4: Plotting these points and connecting them smoothly yields a heart-shaped curve (cardioid) with its cusp at the origin and its maximum distance from the origin at θ = π/2. The shape is a cardioid.

3. Mia is tracking the orbit of a satellite using polar coordinates. The satellite's trajectory is modeled by the polar equation r = 13 + 10cos(θ). If the satellite reaches its closest point to Earth when θ = π, what is that minimum distance r from Earth's center? Answer: 3 Solution: The polar equation is r = 13 + 10cos(θ). At the closest point, θ = π, so cos(π) = -1. Substitute: r = 13 + 10 * (-1) = 13 - 10.
   Full step-by-step solution

   Step 1: The polar equation is r = 13 + 10cos(θ). Step 2: At the closest point, θ = π, so cos(π) = -1. Step 3: Substitute: r = 13 + 10 * (-1) = 13 - 10. Step 4: Calculate: 13 - 10 = 3. The minimum distance from Earth's center is 3 units.

4. Liam is designing a solar panel array that follows a polar equation r(θ) = 3 + 2cos(θ) to maximize sun exposure throughout the day. He needs to calculate the total area covered by the panels during one complete rotation. What is the area enclosed by this polar curve? Answer: 11π Solution: The area enclosed by a polar curve r(θ) from θ = α to θ = β is given by A = (1/2)∫[r(θ)]² dθ. For curves involving cosine terms, trigonometric identities like cos²θ = (1 + cos2θ)/2 can help simplify the integration.
   Full step-by-step solution

   The area enclosed by a polar curve r(θ) from θ = α to θ = β is given by A = (1/2)∫[r(θ)]² dθ. For curves involving cosine terms, trigonometric identities like cos²θ = (1 + cos2θ)/2 can help simplify the integration. This method is commonly used in engineering applications involving rotational symmetry.

5. Find the area enclosed by the polar curve r = 4sin(3θ) in the first quadrant. Express your answer as a simplified multiple of π. Answer: 2π Solution: For polar curves of the form r = asin(nθ) or r = acos(nθ), when n is odd, there are n petals. The area of one petal can be found by integrating from 0 to π/n.
   Full step-by-step solution

   For polar curves of the form r = asin(nθ) or r = acos(nθ), when n is odd, there are n petals. The area of one petal can be found by integrating from 0 to π/n. Since the curve is symmetric, the area in the first quadrant can be determined by considering which portions of which petals fall in that region.

6. ∫(4x³ - 6x² + 2x) dx from 0 to 1 = ? Answer: 0 Solution: Find the antiderivative of 4x³ - 6x² + 2x Antiderivative = (4/4)x⁴ - (6/3)x³ + (2/2)x² = x⁴ - 2x³ + x² Evaluate the antiderivative from 0 to 1 At x = 1: (1)⁴ - 2(1)³ + (1)² = 1 - 2 + 1 = 0 At x = 0: (0)⁴ - 2(0)³ + (0)² = 0 - 0 + 0 = 0 Subtract the lower limit evaluation from the upper limit…
   Full step-by-step solution

   Step 1: Find the antiderivative of 4x³ - 6x² + 2x Antiderivative = (4/4)x⁴ - (6/3)x³ + (2/2)x² = x⁴ - 2x³ + x² Step 2: Evaluate the antiderivative from 0 to 1 At x = 1: (1)⁴ - 2(1)³ + (1)² = 1 - 2 + 1 = 0 At x = 0: (0)⁴ - 2(0)³ + (0)² = 0 - 0 + 0 = 0 Step 3: Subtract the lower limit evaluation from the upper limit evaluation 0 - 0 = 0 The answer is 0.

7. Graph r = 7 + 7sin(θ). Identify the shape and find the maximum value of r. Answer: Cardioid; maximum r = 14 Solution: The equation is r = 7 + 7sin(θ). This is in the form r = a + b sin(θ) with a = 7 and b = 7. Since a = b, the ratio a/b = 1, which produces a cardioid (heart-shaped curve).
   Full step-by-step solution

   Step 1: The equation is r = 7 + 7sin(θ). This is in the form r = a + b sin(θ) with a = 7 and b = 7. Step 2: Since a = b, the ratio a/b = 1, which produces a cardioid (heart-shaped curve). Step 3: The maximum value of sin(θ) is 1, which occurs at θ = π/2. Step 4: Substitute sin(θ) = 1 into the equation: r = 7 + 7(1) = 7 + 7 = 14. Step 5: The minimum value occurs when sin(θ) = -1, giving r = 7 + 7(-1) = 0, which is the cusp of the cardioid. The shape is a cardioid, and the maximum value of r is 14.

8. Liam is designing a roller coaster that follows a spiral path. The track's position in polar coordinates is given by r(θ) = 3θ, where θ is measured in radians. If the roller coaster car starts at θ = 0 and travels to θ = 2π, what is the total distance traveled by the car along this spiral path? Answer: 3π√(1+4π²) + (3/2)ln(2π+√(1+4π²)) Solution: In polar coordinates, the arc length formula accounts for both the changing radius and the angular motion. For a curve defined by r = f(θ), the differential arc length element depends on both the current radius and how quickly the radius changes with respect to the angle.
   Full step-by-step solution

   In polar coordinates, the arc length formula accounts for both the changing radius and the angular motion. For a curve defined by r = f(θ), the differential arc length element depends on both the current radius and how quickly the radius changes with respect to the angle. This creates a Pythagorean-like relationship where the total displacement comes from both the radial and angular components of motion.

9. ∫(x² + 2x - 3) dx from 1 to 4 = ? Answer: 27 Solution: Find the antiderivative of x² + 2x - 3. The antiderivative of x² is (1/3)x³. The antiderivative of 2x is (2/2)x² = x².
   Full step-by-step solution

   Step 1: Find the antiderivative of x² + 2x - 3. The antiderivative of x² is (1/3)x³. The antiderivative of 2x is (2/2)x² = x². The antiderivative of -3 is -3x. So the antiderivative F(x) = (1/3)x³ + x² - 3x. Step 2: Evaluate F(4) - F(1). F(4) = (1/3)(4)³ + (4)² - 3(4) = (1/3)(64) + 16 - 12 = 64/3 + 4 = 64/3 + 12/3 = 76/3 F(1) = (1/3)(1)³ + (1)² - 3(1) = 1/3 + 1 - 3 = 1/3 - 2 = 1/3 - 6/3 = -5/3 Step 3: Calculate F(4) - F(1) = 76/3 - (-5/3) = 76/3 + 5/3 = 81/3 = 27 The answer is 27.