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Home Vector Calculus Surface Area and Surface Integrals Problems  
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1 Find the area of the triangle cut from the plane x + 2y + 4z = 10 by the coordinate planes. 2 Find the area cut from the plane 2x + 4y + z = 0 by the cylinder x^{2} + y^{2} = 1. 3 Find the area of the surface of the paraboloid z = x^{2} + y^{2} below the plane z = 1. 4 Find the area of the surface of the cone below the plane z = 2. 5 Find the surface area of the part of the sphere x^{2} + y^{2} + z^{2} = a^{2} which lies in the first octant; i.e., x ≥ 0, y ≥ 0, z ≥ 0. 6 Find the surface area of the part of the sphere x^{2} + y^{2} + z^{2} = a^{2} which is above the circle x^{2} + y^{2} ≤ b^{2} (b ≤ a). 7 Find the surface area cut from the hyperboloid z = x^{2}  y^{2} by the cylinder x^{2} + y^{2} = a^{2}. 8 Find the area cut from the surface z = xyby the cylinder x^{2} + y^{2} = a^{2}. 9 Find the surface area of the part of the sphere r^{2} + z^{2} = a^{2} above the circle r = a cos θ. 10 Find the surface area of the part of the cone z = cr above the circle r = a cos θ. 11 Find the area of the part of the plane z = ax + by + c over a region D of area A. 12 Find the surface area of the part of the cone over a region D of area A. 13 Find the surface area of the part of the cylinder x^{2} + z^{2} = a^{2} cut out by the cylinder x^{2} + y^{2} ≤ a^{2}. 14 Find the surface area of the part of the cylinder x^{2} + z^{2} = a^{2} above and below the square b ≤ x ≤ b, b ≤ y ≤ b (b ≤ a). 15 Evaluate the surface integral where S is the surface z = x^{2} + y^{2}, 1 ≤ x ≤ 1,  1 ≤ y ≤ 1, oriented with the top side positive. 16 Evaluate the surface integral where S is the surface z = 3x  5y over the rectangle 1 ≤ x ≤ 2, 0 ≤ y ≤ 2, oriented with the top side positive. 17 Evaluate the surface integral where S is the surface z = 1  x^{2}  y^{2}, x^{2} + y^{2} ≤ 1, oriented with the top side positive. 18 Evaluate the surface integral where S is the surface z = x + y^{2} + 2, 0 ≤ x ≤ 1, x ≤ y ≤ 1, oriented with the top side positive. 19 Evaluate the surface integral where S is the surface z = xy, 0 ≤ x ≤ 1,  ,x ≤ y ≤ x, oriented with the top side positive. 20 Evaluate the surface integral where S is the surface , x^{2} + y^{2} ≤ b, oriented with the top side positive (b < a). 21 Show that if S is a horizontal surface z = c over a region D, oriented with the top side positive, then the surface integral over S is


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