Sketches of r = 2+2 cos(0) and r₂ = 0 on the interval [0, 2π] are shown. The intersection point is approximately 0= 1.714 radians. (a) Find the area inside the entire cardioid, accurate to four decimal places. (b) For what 8-value does ri = 0? (c) Find the area inside both curves, accurate to four decimal places. 3
Sketches of r = 2+2 cos(0) and r₂ = 0 on the interval [0, 2π] are shown. The intersection point is approximately 0= 1.714 radians. (a) Find the area inside the entire cardioid, accurate to four decimal places. (b) For what 8-value does ri = 0? (c) Find the area inside both curves, accurate to four decimal places. 3
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 4**
Sketches of \( r_1 = 2 + 2\cos(\theta) \) and \( r_2 = \theta \) on the interval \([0, 2\pi]\) are shown. The intersection point is approximately \( \theta = 1.714 \) radians.
- **(a)** Find the area **inside** the entire cardioid, accurate to four decimal places.
- **(b)** For what \(\theta\)-value does \( r_1 = 0 \)?
- **(c)** Find the area **inside both curves**, accurate to four decimal places.
**Graph Explanation:**
The graph illustrates two polar curves:
1. **Cardioid (blue curve):** Defined by \( r_1 = 2 + 2\cos(\theta) \), which traces a heart-shaped curve symmetric about the horizontal axis.
2. **Line (red curve):** Defined by \( r_2 = \theta \), which is a linear spiral starting from the pole and extending outward.
The plot is set on a polar coordinate system, marked with concentric circles and radial lines for reference. The cardioid intersects the spiral at approximately \( \theta = 1.714 \), which is a significant point for calculations requested in parts (a) and (c).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F14b65783-eb23-4822-951c-32d4f7635fee%2Fb2a5f8d6-91df-4a65-bffd-f029a63ece71%2Fvp5bn5e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 4**
Sketches of \( r_1 = 2 + 2\cos(\theta) \) and \( r_2 = \theta \) on the interval \([0, 2\pi]\) are shown. The intersection point is approximately \( \theta = 1.714 \) radians.
- **(a)** Find the area **inside** the entire cardioid, accurate to four decimal places.
- **(b)** For what \(\theta\)-value does \( r_1 = 0 \)?
- **(c)** Find the area **inside both curves**, accurate to four decimal places.
**Graph Explanation:**
The graph illustrates two polar curves:
1. **Cardioid (blue curve):** Defined by \( r_1 = 2 + 2\cos(\theta) \), which traces a heart-shaped curve symmetric about the horizontal axis.
2. **Line (red curve):** Defined by \( r_2 = \theta \), which is a linear spiral starting from the pole and extending outward.
The plot is set on a polar coordinate system, marked with concentric circles and radial lines for reference. The cardioid intersects the spiral at approximately \( \theta = 1.714 \), which is a significant point for calculations requested in parts (a) and (c).
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