Integral Calculus

Integration

Differentiation gives slopes; integration gives totals.

\int_a^b f(x)\, dx = \text{signed area under } f \text{ on } [a, b].

We need integrals to define probabilities (\int p(x) dx = 1), expectations (\mathbb{E}[X] = \int x\, p(x)\, dx), and multivariate generalizations everywhere a continuous random variable shows up.

Geometric interpretation

Integration starts as area under a nonnegative curve:

Definite integral

Most useful integrals are over an interval [a,b]:

\int_a^b f(x)\,dx.

Geometrically this keeps only the area between the vertical boundaries.

Riemann approximation

Approximate the area by rectangles; refine the partition; the limit is the integral.

'approximation: 0.7944855690002441, truth: [0.804719]'

Multiple integrals

\iint f(x, y)\, dA — total under a 2D surface; iterated integration treats one axis at a time:

# Construct grid and compute function
x, y = tf.meshgrid(tf.linspace(-2., 2., 101), tf.linspace(-2., 2., 101))
z = tf.exp(- x**2 - y**2)

# Plot function
ax = d2l.plt.figure().add_subplot(111, projection='3d')
ax.plot_wireframe(x, y, z)
d2l.plt.xlabel('x')
d2l.plt.ylabel('y')
d2l.plt.xticks([-2, -1, 0, 1, 2])
d2l.plt.yticks([-2, -1, 0, 1, 2])
d2l.set_figsize()
ax.set_xlim(-2, 2)
ax.set_ylim(-2, 2)
ax.set_zlim(0, 1)
ax.dist = 12

Recap

  • Integral = signed area / volume; defined as a limit of Riemann sums.
  • Fundamental theorem of calculus: integral is the inverse of differentiation.
  • Multiple integrals integrate over higher-dimensional regions.
  • Foundation of probability: \int_{\mathcal{X}} p = 1 defines a density; expectation is an integral.