Questions from General Calculus

Q: Use a graph of the vector field F and the curve C

Use a graph of the vector field F and the curve C to guess whether the line integral of F over C is positive, negative, or zero. Then evaluate the line integral. F (x, y) = (x - y) i + xy j, C is the...

Q: Let S be a smooth parametric surface and let P be a

Let S be a smooth parametric surface and let P be a point such that each line that starts at P intersects S at most once. The solid angle Ω (S) subtended by S at P is the set of li...

Q: Use a graph of the vector field F and the curve C

Use a graph of the vector field F and the curve C to guess whether the line integral of F over C is positive, negative, or zero. Then evaluate the line integral. C is the parabola y = 1 + x2 from (-...

Q: Find the area of the surface. The part of the

Find the area of the surface. The part of the surface x = z2 + y that lies between the planes y = 0, y = 2, z = 0, and z = 2

Q: Find the area of the surface. The part of the

Find the area of the surface. The part of the paraboloid y = x2 + z2 that lies within the cylinder x2 + z2 = 16

Q: The base of a circular fence with radius 10 m is given

The base of a circular fence with radius 10 m is given by x = 10 cos t, y = 10 sin t. The height of the fence at position (x, y) is given by the function h (x, y) = 4 + 0.01 (x2 - y2), so the height v...

Q: Suppose f is a continuous function defined on a rectangle R =

Suppose f is a continuous function defined on a rectangle R = [a, b] × [c, d]. (a). Write an expression for a double Riemann sum of f. If f (x, y) > 0, what does the sum represent? (b). Write the defi...

Q: Find the area of the surface. The part of the

Find the area of the surface. The part of the sphere x2 + y2 + z2 = b2 that lies inside the cylinder x2 + y2 = a2, where 0 < a < b

Q: An object moves along the curve C shown in the figure from

An object moves along the curve C shown in the figure from (1, 2) to (9, 8). The lengths of the vectors in the force field F are measured in newtons by the scales on the axes. Estimate the work done b...

Q: Experiments show that a steady current I in a long wire produces

Experiments show that a steady current I in a long wire produces a magnetic field B that is tangent to any circle that lies in the plane perpendicular to the wire and whose center is the axis of the w...