Converting to Polar Coordinates:
In Exercises 17–26, evaluate the iterated
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Calculus (MindTap Course List)
- Find ʃR (x2 + y2) dA where R is the region 4 ≤ (x2 + y2)≤ 9. Hint: Use polar coordinates.arrow_forwardConverting to a polar integral Integrate ƒ(x, y) = [ln (x2 + y2 ) ]/sqrt(x2 + y2) over the region 1<= x2 + y2<= e.arrow_forwardTranslation of text in image: where R is the region of the XY plane, given by R = R1 ∪ R2, and represented in the attached graph When transforming the previous integral applying the change of variable to polar coordinates, we obtain:arrow_forward
- A. State the Fundamental Theorem of Calculus for Line Integrals. B. Let f(x, y, z) = x^2 + 2y^2 + 3z^2 and F = grad f. Find the line integral of F along the line C with parametric equations x = 1 + t, y = 1 + 2t, z = 1 + 3t, 0 ≤ t ≤ 1. You must compute the line integral directly by using the given parametrization. C. Check your answer in Part B by using the Fundamental Theorem of Calculus for Line Integrals.arrow_forwardA. State the F undamental Theorem of Calculus for Line Integrals. B. Let f(x, y, z) = xy + 2yz + 3zx and F = grad f. Find the line integral of F along the line C with parametric equations x = t, y = t, z = 3t, 0 ≤ t ≤ 1. You must compute the line integral directly by using the given parametrization. C. Check your answer in Part B by using the Fundamental Theorem of Calculus for Line Integrals.arrow_forwardDouble integral to line integral Use the flux form of Green’sTheorem to evaluate ∫∫R (2xy + 4y3) dA, where R is the trianglewith vertices (0, 0), (1, 0), and (0, 1).arrow_forward
- find the centroid,using polar coordinates the first quadrant area bounded by the curve r=a sin 2 θarrow_forwardShow all solution. Include the graph and figures. Evaluate the iterated double integral using polar coordinates.arrow_forwarda) Sketch the region of integration b) Express the region in polar coordinates c) Write an equivalent double integral in polar coordinatesarrow_forward
- Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. Cartesian in images* I need the answer for the boxes 1, 2, and 3 and the value of the doublearrow_forwarda.Find the centroid of the region in the polar coordinate plane that lies inside the cardioid r = 1 + cos u and outside the circle r = 1. b. Sketch the region and show the centroid in your sketch.arrow_forwardStokes’ Theorem for evaluating surface integrals Evaluate the line integral in Stokes’ Theorem to determine the value of the surface integral ∫∫S (∇ x F) ⋅ n dS. Assume n points in an upward direction. F = ⟨4x, -8z, 4y⟩; S is the part of the paraboloidz = 1 - 2x2 - 3y2 that lies within the paraboloid z = 2x2 + y2 .arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning