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.
Integration starts as area under a nonnegative curve:
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.
Approximate the area by rectangles; refine the partition; the limit is the integral.
\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 = np.meshgrid(np.linspace(-2, 2, 101), np.linspace(-2, 2, 101),
indexing='ij')
z = np.exp(- x**2 - y**2)
# Plot function
ax = d2l.plt.figure().add_subplot(111, projection='3d')
ax.plot_wireframe(x.asnumpy(), y.asnumpy(), z.asnumpy())
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