1. Circumference of a circle: Consider the polar equation r = 2Rsine. Sketch the graph, identify the shape and main characteristics, and note the limits for 0. Use the polar arc length formula to derive the formula for circumference of a circle.
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- Find the area inside one petal of the polar equationFind the area A of the sector of a circle of radius 10 inches formed by the central angle 12 radian.2. Area of a circle: Consider the polar equation r = 2Rsine. Sketch the graph and note the limits for 0. Use the polar area formula to derive the formula for area of a circle.
- Derive CRE in polar coordinates.Show that r = a cos θ + b sin θ is the equation of a circle passing through the origin. Express the radius and center (in rectangular coordinates) in terms of a and b and write down the equation in rectangular coordinates.Find cose, where 0 is the angle shown. Give an exact value, not a decimal approximation.
- Mackenzie is skiing along a circular ski trail that has a radius 3.2 km long. She starts at the 3-o'clock position and travels in the CCW direction. Mackenzie stops skiing when she is -0.57 km to the right and 3.149 km above the center of the ski trail. Imagine an angle with its vertex at the center of the ski trail that subtends Mackenzie's path. a. How many radians has the angle swept out since Mackenzie started skiing? radians Preview b. How many km has Mackenzie skied since she started skiing? km PreviewWhich of the following integrals represent the arc lengthof the curve y = sinx ,0 S xS (a) 1+sin2 (x) dx (b) V1- cos²(x) dx (c) V1+ cos² (x) dx (d) V1- sin? (x) dx (e) None aFind the area of the sector of a circle with radius 4 kilometers formed by a central angle of 3π/2 radians. Give your answer in terms of pi.
- Find the exact length of the polar curve.The polar equation r cose = 3 represents: %3D (a) Cardioid (b) Vertical line (c)Horizontal line (d) Circle (e) NoneConvert the following polar coordinates to Cartesian coordinates. Please use 6 significant digits in your calculations. Remember that polar coordinates use radians to measure θ. a)(r, θ) = (7.2, 1.3π) b) (r, θ) = (6.4, 0.81π) c) (r, θ) = (−1.5, −0.71π) d) (r, θ) = (−3.9, 0.73π)