Questions from General Calculus

Q: The figure shows the surface created when the cylinder y2 + z2

The figure shows the surface created when the cylinder y2 + z2 = 1 intersects the cylinder x2 + z2 = 1. Find the area of this surface.

Q: Find the area of the part of the sphere x2 + y2

Find the area of the part of the sphere x2 + y2 + z2 = a2 that lies inside the cylinder x2 + y2 = ax.

Q: At time t = 1, a particle is located at position

At time t = 1, a particle is located at position (1, 3). If it moves in a velocity field F (x, y) = 〈xy - 2, y^2 - 10〉 find its approximate location at time t = 1.05.

Q: Find the positively oriented simple closed curve C for which the value

Find the positively oriented simple closed curve C for which the value of the line integral ∫C (y3 – y) dx - 2x3 dy is a maximum.

Q: The flow lines (or streamlines) of a vector field are

The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus, the vectors in a vector field are tangent to the flow lines...

Q: (a). Sketch the vector field F (x, y

(a). Sketch the vector field F (x, y) = i + x j and then sketch some flow lines. What shape do these flow lines appear to have? (b). If parametric equations of the flow lines are x = x (t), y = y (t),...

Q: What do the following statements mean? (a). f

What do the following statements mean? (a). f has a local maximum at (a, b). (b). f has an absolute maximum at (a, b). (c). f has a local minimum at (a, b). (d). f has an absolute minimum at (a, b). (...

Q: Explain how the method of Lagrange multipliers works in finding the extreme

Explain how the method of Lagrange multipliers works in finding the extreme values of f (x, y, z) subject to the constraint g (x, y, z) = k. What if there is a second constraint h (x, y, z) = c?

Q: (a). Write formulas for the unit normal and binormal vectors

(a). Write formulas for the unit normal and binormal vectors of a smooth space curve r (t). (b). What is the normal plane of a curve at a point? What is the osculating plane? What is the osculating ci...

Q: If F (x, y) = sin y i +

If F (x, y) = sin y i + (1 + x cos y) j, use a plot to guess whether F is conservative. Then determine whether your guess is correct.