Evaluating a Double
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Calculus: Early Transcendental Functions
- Converting to a polar integral Integrate ƒ(x, y) = [ln (x2 + y2 ) ]/sqrt(x2 + y2) over the region 1<= x2 + y2<= e.arrow_forwardConsider the double integral ∫ ∫ D (x^2 + y^2 ) dA, where D is the triangular region with vertices (0, 0), (0, 1), and (1, 2). A. Describe D as a vertically simple region and express the double integral as an iterated integral. B. Describe D as a horizontally simple region and express the double integral as an iterated integral. C. Evaluate the iterated integral found either in part A or in part B.arrow_forwardFinding the Volume of a Solid In Exercises 17-20, find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 4.y =1/2x3, y = 4, x = 0arrow_forward
- M3. Subject :- Advance math The surface surrounding the region bounded by the x2 + y2 = 4 cylinder and the z = 0 and z = 3 planes is S, the normal directed outward. Accordingly, (x, y, z) = x i + y j + z k using the Divergens theorem to calculate the integral of F. n dSarrow_forward(a) Find the Jacobian of the transformation x = u, y = uv(b) Sketch the region G: 1 ≤ u ≤ 2, 1 ≤ uv ≤ 2 in the uv-plane(c) Using the above transformation, transform the integral (picture included of integral) into an integral over G, and evaluate both integrals.arrow_forwarduse a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. F(x, y, z) = x4 + y2 + z2 over the solid sphere x2 + y2+z2 <= 1arrow_forward
- Check Stokes' Theorem, evaluating the two integrals of the statement, to F(x, y, z) = (y, −x, 0), the paraboloid S : z = x2 + y2, with 0 ≤ z ≤ 1, and n pointing out of S. Answer is 1/2arrow_forwardEvaluate the line integral using Green's Theorem and check the answer by evaluating it directly. ∮C4 y2dx+6 x2dy∮C4 y2dx+6 x2dy, where CC is the square with vertices (0,0)(0,0), (3,0)(3,0), (3,3)(3,3), and (0,3)(0,3) oriented counterclockwise. ∮C4 y2dx+6 x2dy=arrow_forwarda.) Find the centroid of the first quadrant area bounded by x = y and x2 − y = 0. b.) Find the centroid of the area bounded by y2 − x = 0 and x2 + 8y = 0.arrow_forward
- 34) The figure shows the region of integration for the integral. Rewrite this itegral as an equivalent iterated integral in the five other orders. ∫0 to 1 ∫0 to (1-x^2) ∫0 to (1-x) f(x, y, z) dydzdxarrow_forwardWrite a double integral that represents the surface area of z = f (x, y) that lies above the region R. Use a computer algebra system to evaluate the double integral. f(x, y) = 2y + x2, R: triangle with vertices (0, 0), (1, 0), (1, 1)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 = ⟨y, z - x, -y⟩; S is the part of the paraboloidz = 2 - x2 - 2y2 that lies within the cylinder x2 + y2 = 1.arrow_forward
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