S The graphs of the polar curves r1(0) = cos 0 and r2(0) = 2 sin 0 are shown in the figure above. (a) Let S be the region that is inside the graph of r1 and also inside the graph of r2. Find the area of S.
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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).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.Sketch r = 4 + 8 cos θ on the image below and find the area outside the inner loop of the curve using polar coordinates integration.
- Find an iterated integral in polar coordinates that represents the area of the given region in the polar plane and then evaluate the integral. One loop of the curve r = 4 sin3θ.The parametric equations x = 4 - 5 sin 3t and y = 3 + 5 cos 3t for 0 ≤ t ≤ π/2 represent a curve C. Draw the curve C, indicating the direction of increasing t and then find the length L of the curve C, using an appropriate integral.Use the integration capabilities of a graphing utility to approximate the area of the region bounded by the graph of the polar equation. r = 2/(7 − 6 sin)
- A. Find the arc length of the curve c(t) = (x(t),y(t)) = (sin(3t), cos(3t)) for 0 les or equal to t and less or equal to π B. Express the Cartesian coordinate (2, 3) in polar coordinates in at least three different ways C. Consider the four petaled rose r = sin(2θ). Find the area of one leaf, then prove that the total area of the rose is equal toone-half the area of the circumscribed circle.Find the area in the first quadrant inside the curve r = 4 + cos 2θ but outside the curve r = 3 cos 2θ (see figure), after first finding the intersection points of the two curves.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.
- 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.Graph (either by hand or desmos) the polar curves r = 2 andr = 4 − 4 sin θ. Use a double integral to find the area inside thecircle, but outside of the cardioid.II. Consider the circle C1 : r = 1 and the roses C2 : r = cos 2θ and C3 : r = 2 cos 2θ, each of which is symmetric with respect to the polar axis, the π/2-axis, and the origin, as shown on the image. 1. Find polar coordinates (r, θ) for the intersection A of C1 and C3, where r, θ > 0. 2. Set-up (do not evaluate) a sum of three definite integrals that give the perimeter of the yellow-shaded region inside both C1 and C3 but outside C2. 3. Find the area of the unshaded region inside C3 but outside C1.