Questions from General Calculus

Q: Let F be an inverse square field, that is, F

Let F be an inverse square field, that is, F (r) = cr/|r |3 for some constant c, where r = x i + y j + z k. Show that the flux of F across a sphere S with center the origin is independent of the radiu...

Q: If C is a smooth curve given by a vector function r

If C is a smooth curve given by a vector function r (t), a

Q: If the equation of a surface S is z = f (

If the equation of a surface S is z = f (x, y), where x2 + y2 < R2, and you know that | fx | < 1 and | fy | < 1, what can you say about A (S)?

Q: Find the area of the surface correct to four decimal places by

Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral. The part of the surface z = cos (x2...

Q: Find the area of the surface correct to four decimal places by

Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral. The part of the surface z = ln (x2...

Q: Find, to four decimal places, the area of the part

Find, to four decimal places, the area of the part of the surface z = (1 + x2)/ (1 + y2) that lies above the square |x | + |y | < 1. Illustrate by graphing this part of the surface.

Q: (a). Use the Midpoint Rule for double integrals (see

(a). Use the Midpoint Rule for double integrals (see Section 15.1) with six squares to estimate the area of the surface z = 1/ (1 + x2 + y2), 0 < x < 6, 0 < y < 4. (b). Use a computer algebra system t...

Q: Let F (x, y) = (2x3 + 2xy2

Let F (x, y) = (2x3 + 2xy2 - 2y) i + (2y3 + 2x2y + 2x) j x2 + y2 Evaluate ∫C F ∙ dr, where C is shown in the figure.

Q: Find the area of the surface with vector equation r (u

Find the area of the surface with vector equation r (u, v) = 〈cos3u cos3v, sin3u cos3v, sin3v〉, 0 < u < u, 0 < v < 2π. State your answer corrects to four decimal places.

Q: Find the exact area of the surface z = 1 + 2x

Find the exact area of the surface z = 1 + 2x + 3y + 4y2, 1 < x < 4, 0 < y < 1.