A Complete Guide to Integral Calculus (Advanced Calculus)

Explore the element of area for two-variable functions in Cartesian coordinates and polar coordinates. Derive da = dx dy and da = r dr d theta using r and theta.

derive the area element in generalized coordinates using the jacobian of x and y with respect to u and v, and illustrate with cartesian and polar coordinates.

Transform double integrals by converting x,y to u,v using the Jacobian and generalized element of area, express the integrand as F(u,v), and adjust the region.

Transform coordinates to convert complex double-integral regions into a rectangle in (u, v) for the moment of inertia example, then use a second transformation to simplify the exponential integrand.

Examine geometric and physical applications of double integrals, including area, volume under a surface, volumes by revolution, centroid, mass, average value, and moment of inertia.

Explore the gamma function defined by the integral from zero to infinity of e^{-x} x^{n-1} dx, with recursion gamma(n+1)=n gamma(n) and gamma(1)=1, yielding gamma(n+1)=n! for positive integers.

Compute gamma of one half by transforming the gamma integral with substitutions and polar coordinates to obtain gamma(1/2)=sqrt(pi), and apply the recursion gamma(n+1)=n gamma(n) to reach gamma(3/2) and beyond.

Derive an equation to evaluate integrals involving products of sine and cosine using the gamma function, and relate the integral to gamma(n) and gamma(m) via polar coordinates.

Study how to evaluate integrals with the gamma function through three examples, handling cosine and sine powers, quadrant rules, and the zero result before previewing the Laplace transform.

Explore the Laplace transform as an integral transform with kernels e^{-st}, derive t^n, e^{at}, cosh, and cos transforms, and learn existence conditions via piecewise continuity and exponential order.

Evaluate triple integrals in cylindrical coordinates to compute the hurricane's total kinetic energy from density delta0 e^{-z/h} and theta-directed velocity, identifying max velocity at r=a, z=0.

Learn to transform coordinates in triple integrals by expressing x, y, z as functions of u, v, w using the Jacobian determinant and explore cylindrical coordinates with r, theta, z.

Set up triple integrals in cylindrical coordinates to find the volume inside a sphere of radius a but outside a cylinder of radius b, yielding 4 pi/3 (a^2 - b^2)^{3/2}.

Compute the volume of a sphere of radius two cut by the cylinder x^2+y^2=2x using cylindrical coordinates with theta in [-pi/2, pi/2] and r in [0, 2 cos theta].

Compute the mass of the smaller part of a sphere cut by z = 1/2 with density, via a triple integral in cylindrical coordinates, switching the order to yield pi/4.

Introduce spherical coordinates with rho, phi, theta, where rho is distance from the origin; derive x, y, z relations and the jacobian rho^2 sin phi for triple integrals.

Compute the center of mass of a constant-density ice cream cone in spherical coordinates using triple integrals; show x and y vanish by symmetry and z equals (3a/8)(1+cos alpha).

Compute the atmosphere's mass with triple integrals in spherical coordinates. Model density as delta0 e^{-k(rho - a)} and integrate rho from a to infinity, yielding a closed-form expression.

Compute the normalization constant k for the hydrogen atom wave function by enforcing the triple integral of |psi|^2 in spherical coordinates equals one, yielding k = sqrt(c^3/pi).

Learn to compute the moment of inertia and radius of gyration for a solid homogeneous sphere of constant density using triple integrals in spherical coordinates.

Revisit the mass computation in spherical coordinates for a spherical cap defined by z=1/2, setting rho, phi, theta bounds and applying delta with the jacobian to verify pi/4.

Revisit the volume inside the sphere but outside the cylinder using spherical coordinates, deriving bounds and showing the result equals (4π/3)(a^2−b^2)^{3/2}, the volume of a sphere with radius sqrt(a^2−b^2).

Define vector functions r(u,v) to describe surfaces like spheres, cylinders, and cones in Cartesian, cylindrical, and spherical coordinates, and compute surface area via |∂r/∂u × ∂r/∂v|.

Derive surface area in cartesian coordinates by modeling z=f(x,y), computing r_x cross r_y, and integrating the magnitude sqrt(f_x^2+f_y^2+1) over the xy-projection.

Compute the sphere's surface area in Cartesian coordinates by expressing z as a function of x and y and using symmetry and polar coordinates to obtain 4 pi a^2.

Show how cartesian surface integrals locate the cylinder by completing the square and use polar bounds to compute the sphere portion, yielding 2 pi minus 4.

Derive the surface area of a flat cone in Cartesian coordinates by relating z to x and y, using similar triangles, converting to polar coordinates, and obtaining pi b L.

Compute surface areas in spherical coordinates by applying the cross product magnitude of r_phi and r_theta, yielding the integrand a^2 sin phi and setting bounds for phi and theta.

Shows how to compute force on a point charge by a uniformly charged hollow sphere using surface integrals in spherical coordinates, applying Coulomb's law for outside, inside, and on-surface cases.

Demonstrates integrating gravity from a solid homogeneous sphere in spherical coordinates, yielding outside result F = G M_sphere m / b^2 and linking to the hollow sphere case.