Using Green's Theorem, evaluate |(sin z – y) da + (2³ + e") dy where C'is the circle x2 + y = 4. Enter your answer as a decimal rounded to four decimal places. In your written work, make sure you have the exact answer. Hint: You may want to use polar coordinates in your integral at some point.
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- Use Green’s theorem to evaluate 3 3C∫ y dx − x dy where C is the curve shown:where r1=3 and r2=5 units. Use polar coordinates to evaluate the resulting double integral.Use the integration capabilities of a graphing utility to approximate the area of the surface formed by revolving the polar equation over the given interval about the polar axis. r = 4 cos 2θ, [0, π/ 4]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.
- Use double integral to find the volume of the cylinder-like object that exists above the first quadrant on the xy plane and below the plane z = 10. The cross section of this "cylinder" on the xy plane is given by the polar equation r = 50 sin (2θ)Consider the parametric equations below. x = t sin(t), y = t cos(t), 0 ≤ t ≤ π/6 Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis.Find the area of the surface obtained by rotating the curve y = cos 2x, x is an element of [ 0, pi/6 ] about the x-axis.
- consider the function in polar coordinates f(r, θ) = (cos(2θ) − r) sin(πr) on the region inside the circle r = 1. Set up the double integral in polarcoordinates to find the volume of the region below f(r, θ) and above the xy-plane.Find the area enclosed by the simple closed curve given by the parametric equations, x = 2cost + cos2t and y = 2sint − sin2t for t ∈ [0,2π]. The graph is shown below.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.
- 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.Consider the parametric equations x = a cos3 t and y = a sin3 t with 0 ≤ t ≤ π. Find the surface area of the solid obtained by rotating the region about the x-axis.