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_forwardM3. 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) Sketch the region of integration R in the xy - plane and sketch the region G in the uv - plane using the coordinate transformation x = 2u and y = 2u + 4v.arrow_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_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_forward34) 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_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_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_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_forward
- Six orderings Let D be the solid in the first octant bounded bythe planes y = 0, z = 0, and y = x, and the cylinder 4x2 + z2 = 4.Write the triple integral of ƒ(x, y, z) over D in the given order of integration. dy dz dxarrow_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_forwardSix orderings Let D be the solid in the first octant bounded bythe planes y = 0, z = 0, and y = x, and the cylinder 4x2 + z2 = 4.Write the triple integral of ƒ(x, y, z) over D in the given order of integration. dx dz dyarrow_forward
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