Questions from General Calculus

Q: Use Exercise 22 to find the centroid of a quarter-circular

Use Exercise 22 to find the centroid of a quarter-circular region of radius a. Exercise 22: Let D be a region bounded by a simple closed path C in the xy-plane. Use Green’s Theorem...

Q: Use Exercise 22 to find the centroid of the triangle with vertices

Use Exercise 22 to find the centroid of the triangle with vertices (0, 0), (a, 0), and (a, b), where a > 0 and b > 0. Exercise 22: Let D be a region bounded by a simple closed path C in the xy...

Q: A plane lamina with constant density ρ (x, y)

A plane lamina with constant density ρ (x, y) = ρ occupies a region in the xy-plane bounded by a simple closed path C. Show that its moments of inertia about the axes are

Q: Use Exercise 25 to find the moment of inertia of a circular

Use Exercise 25 to find the moment of inertia of a circular disk of radius a with constant density ρ about a diameter. (Compare with Example 15.4.4.) Exercise 25: A plane lamina with cons...

Q: Plot the gradient vector field of f together with a contour map

Plot the gradient vector field of f together with a contour map of f. Explain how they are related to each other. f (x, y) = ln (1 + x2 + 2y2)

Q: Calculate ∫C F ∙ dr, where F (x,

Calculate ∫C F ∙ dr, where F (x, y) = 〈x2 + y, 3x - y2〉 and C is the positively oriented boundary curve of a region D that has area 6.

Q: If F and G are vector fields whose component functions have continuous

If F and G are vector fields whose component functions have continuous first partial derivatives, show that curl (F × G) = F div G - G div F + (G ∙ ∇) F – (F ∙∇) G

Q: Evaluate the line integral by two methods: (a) directly

Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem. ∮C xy dx + x2y3 dy, C is the triangle with vertices (0, 0), (1, 0), and (1, 2)

Q: Complete the proof of the special case of Green’s Theorem by proving

Complete the proof of the special case of Green’s Theorem by proving Equation 3.

Q: What does Clairaut’s Theorem say?

What does Clairaut’s Theorem say?