R The graphs of the polar curves r= 2 and r= 3+ 2cos e are shown in the figure above. The curves intersect when 6= and e Find the area of the shaded region.
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- Find the exact length of the polar curve r=e2(theta) , where 0 is less than or equal too theta is less than or equal too ln(3).The graphs of the polar curves r=5 and r=5+2sin(3ø) are shown in the figure for 0 is less than or equal to ø is less than or equal to 2 pi. A particle is moving along the curve r=5+2sin (3ø) so that dø/dt=5 for all times t is greater than or equal to 0 find the value of dr/dt at ø =pi/96. Consider the curves: r = 2 + cos 2θ and r = 2 + sin 2θ, a) Sketch a graph of both curves on the polar graph provided. b) Find all points of intersection. c) Find the total area inside r = 2 + cos (2θ) and outside r = 2 + sin (2θ)
- 2. Consider the polar curves r = 4 - 2cosθ and r = 2 + 2cosθ. In this problem, we want to find the area of A, B, and C pictured below. (c) B is the area inside both r = 2 + 2cosθ and r = 4 - 2cosθ. Find the area of B. (Hint: What happens at the angle where the two polar curves intersect? Your answer should involve a sum of two polar integrals.)2. Consider the polar curves r = 4 - 2cosθ and r = 2 + 2cosθ. In this problem, we want to find the area of A, B, and C pictured below. (d) C is the area inside r = 4 - 2cosθ but outside r = 2 + 2cosθ. Find the area of C. (Hint: The inner and outer polar curves switch for C in comparison to A.)The graphs of the polar curves r=5 and r=5+2sin(3ø) are shown in the figure for 0 is less than or equal to ø is less than or equal to 2 pi. Write the expressions for dx/dø and dy/dø for the curve r=5+2sin(3ø).
- 4. Find the exact length of the polar curve r = e2θ, 0 ≤ θ ≤ 2π.Given the graphs of polar equations: r=1 and r=−2cosθ, (a) Set up the integral that gives the area of the shaded region. (b) Evaluate the area of the region showing all your steps.6. Consider the curves: r = 2 + cos2? and r = 2 + sin2?a) Sketch a graph of both curves on the polar graph provided.b) Find all points of intersection.c) Find the total area inside r = 2 + cos (2?) and outside r = 2 + sin (2?)
- Graph (either by hand or desmos) the polar curves r = 3 andr = 3 + 3 cos θ. Use a double integral to find the area inside thecircle, but outside of the cardioid.The graphs of the polar curves r=5 and r=5+2sin(3ø) are shown in the figure for 0 is less than or equal to ø is less than or equal to 2 pi. Let R be the area of the shaded region inside the graph of r=5 and inside the graph of r=5+2sin(3ø) from 0 is less than or equal to ø is less than or equal to pi. Write an integral expression for the area of R.Consider polar curves C1 : r = −3 sin(2θ) and C2 : r = 3 sin θ.Set up the definite integral for the perimeter and area of the region outside C1 but inside C2. See graph below