Find the surface area of the part of the hyperbolic paraboloid z=x^(2)-y^(2) that lies within the cylinder x^(2)+y^(2)=1 in the first octant

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Find the surface area of the part of the hyperbolic paraboloid  z=x^(2)-y^(2) that lies within the cylinder  x^(2)+y^(2)=1 in the first octant

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Identify Region:  Identify the region of integration. Since we are dealing with the first octant and the cylinder x^2 + y^2 = 1, the limits for x and y are both from 0 to 1, and x^2 + y^2 ≤ 1.Set Up Integral:  Set up the double integral for the surface area. The surface area S of z = f(x, y) over a region D is given by the integral ∫∫_D sqrt(1 + (f_x)^2 + (f_y)^2) dA, where f_x and f_y are partial derivatives of f with respect to x and y, respectively.Calculate Derivatives:  Calculate the partial derivatives. For z = x^2 - y^2, f_x = 2x and f_y = -2y. Then, (f_x)^2 = (2x)^2 = 4x^2 and (f_y)^2 = (-2y)^2 = 4y^2.Substitute in Formula:  Substitute into the surface area formula. The integrand becomes sqrt(1 + 4x^2 + 4y^2). The integral for the surface area is ∫∫_D sqrt(1 + 4x^2 + 4y^2) dA.Convert to Polar:  Convert to polar coordinates for easier integration. x = r cos(θ), y = r sin(θ), and dA = r dr dθ. The limits for r are from 0 to 1, and for θ from 0 to π/2 since we are in the first octant.Substitute Polar Coordinates:  Substitute polar coordinates into the integral. The integral becomes ∫ from 0 to π/2 ∫ from 0 to 1 sqrt(1 + 4r^2) r dr dθ.Integrate with Respect to r:  Perform the integration with respect to r first. ∫ from 0 to 1 r sqrt(1 + 4r^2) dr. Let u = 1 + 4r^2, then du = 8r dr, and r dr = du/8. The limits change to u = 1 when r = 0 and u = 5 when r = 1.Continue Integration:  Continue the integration. The integral becomes 1/8 ∫ from 1 to 5 sqrt(u) du. This integral evaluates to 1/8 * (2/3) * (u^(3/2)) from 1 to 5.Evaluate Integral:  Evaluate the integral and multiply by the angular range. The integral of u^(3/2) from 1 to 5 is (2/3) * (5^(3/2) - 1^(3/2)). Calculate 5^(3/2) = 11.1803 and 1^(3/2) = 1. The result is (2/3) * (11.1803 - 1) = 6.7869. Then, 1/8 * 6.7869 = 0.8484. Multiply by the angular range π/2 = 1.5708. Final result is 1.5708 * 0.8484 = 1.332.

Find the surface area of the part of the hyperbolic paraboloid $z=x^{2}-y^{2}$ that lies within the cylinder $x^{2}+y^{2}=1$ in the first octant

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Find the surface area of the part of the hyperbolic paraboloid z=x2−y2 z=x^{2}-y^{2} z=x2−y2 that lies within the cylinder x2+y2=1 x^{2}+y^{2}=1 x2+y2=1 in the first octant

Find the surface area of the part of the hyperbolic paraboloid $z=x^{2}-y^{2}$ that lies within the cylinder $x^{2}+y^{2}=1$ in the first octant