Questions from General Calculus

Q: Use spherical coordinates. Evaluate ∭E y2 dV, where

Use spherical coordinates. Evaluate ∭E y2 dV, where E is the solid hemisphere x2 + y2 + z2 < 9, y > 0.

Q: Use Lagrange multipliers to give an alternate solution to the indicated exercise

Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7. Exercise 44 14.7 Exercise 44: Find the points on the surface y2 = 9 + xz that are closest to the ori...

Q: Evaluate the integral by making an appropriate change of variables.

Evaluate the integral by making an appropriate change of variables. ∬R cos (y-x)/(y+x) dA, where R is the trapezoidal region with vertices (1, 0), (2, 0), (0, 2), and (0, 1)

Q: Evaluate the integral by making an appropriate change of variables.

Evaluate the integral by making an appropriate change of variables. ∬R sin (9x2 + 4y2) dA, where R is the region in the first quadrant bounded by the ellipse 9x2 + 4y2 = 1

Q: Use spherical coordinates. Find the volume of the part of

Use spherical coordinates. Find the volume of the part of the ball ρ < a that lies between the cones φ = π/6 and φ = π/3.

Q: Let f be continuous on [0, 1] and let

Let f be continuous on [0, 1] and let R be the triangular region with vertices (0, 0), (1, 0), and (0, 1). Show that

Q: Express the integral ∭E f (x, y, z

Express the integral ∭E f (x, y, z) dV as an iterated integral in six different ways, where E is the solid bounded by the given surfaces. y = 4 - x2 - 4z2, y = 0

Q: Use spherical coordinates. Find the volume of the solid that

Use spherical coordinates. Find the volume of the solid that lies within the sphere x2 + y2 + z2 = 4, above the xy-plane, and below the cone z = √(x^2+y^2 ).

Q: Use spherical coordinates. (a). Find the centroid of

Use spherical coordinates. (a). Find the centroid of the solid in Example 4. (Assume constant density K.) (b). Find the moment of inertia about the z-axis for this solid.

Q: Use spherical coordinates. Let H be a solid hemisphere of

Use spherical coordinates. Let H be a solid hemisphere of radius a whose density at any point is proportional to its distance from the center of the base. (a). Find the mass of H. (b). Find the center...