Triple integrals in spherical coordinates describe each point by its distance ρ from the origin and two angles, θ and φ. You write x = ρ sin φ cos θ, y = ρ sin φ sin θ and z = ρ cos φ, and replace dV with ρ² sin φ dρ dφ dθ.
Triple integrals in cylindrical coordinates use r, θ and z in place of x, y and z. You write x = r cos θ and y = r sin θ, keep z, and replace dV with r dz dr dθ. This makes integrals over cylinders, cones and paraboloids much easier.
A triple integral adds up the values of f(x, y, z) over a solid region E in space. You evaluate triple integrals as iterated integrals, one variable at a time from the inside out, usually z first. With f = 1, the triple integral gives the volume of E. With a density f, it gives the mass.
Applications of double integrals add up a quantity spread over a flat region. With a density ρ(x, y), the double integral of ρ gives the mass of a thin plate. Dividing its moments by the mass gives the center of mass. The double integral of √(1 + fx² + fy²) gives the area of a surface z = f(x, y).
Double integrals in polar coordinates replace x and y with r and θ, using x = r cos θ and y = r sin θ. The area piece becomes dA = r dr dθ, with an extra factor of r. Use polar coordinates when the region is a disk, a ring or a sector, or when the integrand contains x² + y².
Double integrals over general regions handle regions with curved or slanted edges. For a region where y runs from a bottom curve g₁(x) to a top curve g₂(x) as x goes from a to b, integrate y first between those curves, then x from a to b. The inner limits can be functions; the outer limits must be numbers.
A double integral adds up the values of f(x, y) over a region R in the plane. Over a rectangle R = [a, b] × [c, d], you evaluate double integrals as iterated integrals: integrate with respect to one variable while holding the other fixed, then integrate that result. When f ≥ 0, the double integral is the volume under the surface.
Lagrange multipliers find the maximum or minimum of f(x, y) subject to a constraint g(x, y) = k. At the best points, the gradients are parallel: ∇f = λ∇g for some number λ. Solve these equations together with g = k, then compare the values of f at every solution to pick the largest and smallest.
Local maximums and minimums of f(x, y) happen only at critical points, where f_x = 0 and f_y = 0. The second derivative test uses D = f_xx f_yy − (f_xy)²: if D > 0, it is a minimum when f_xx > 0 and a maximum when f_xx < 0. If D < 0, it is a saddle point.
The gradient of f(x, y) is the vector ∇f = ⟨f_x, f_y⟩. The directional derivative in the direction of a unit vector u is D_u f = ∇f · u, the rate f changes as you move that way. The gradient points in the direction of steepest increase, and its length is that greatest rate.