Integral Calculus

Besides the examples on this page, please see the discussion in BasicCalculus.

Definite and Indefinite Integrals

SAGE can compute both definite integrals like \int_0^1 \frac{dx}{x^3+1} and indefinite integrals such as \int \frac{dx}{x^3+1}:

sage: print integrate(1/(x^3+1),x)
                                         2 x - 1
                       2            atan(-------)
                  log(x  - x + 1)        sqrt(3)    log(x + 1)
                - --------------- + ------------- + ----------
                         6             sqrt(3)          3
sage: integrate(1/(x^3+1), x, 0, 1)
(6*log(2) + sqrt(3)*pi)/18 + sqrt(3)*pi/18

More examples:

sage: integrate(1/x^2, x, 1, infinity)
1
sage: f = x^3 
sage: f.integral()
x^4/4
sage: integral(x^3,x)
x^4/4
sage: f = x*sin(x^2)
sage: integral(f,x)
-cos(x^2)/2

The Definite Integral

o The definition of area under curve o Relation between velocity and area o Definition of Integral o The Fundamental Theorem of Calculus

Indefinite Integrals and Change

o Indefinite Integrals o Physical Intuition

Substitution and Symmetry

o The Substitution Rule o The Substitution Rule for Definite Integrals o Symmetry

Applications to Areas, Volume, and Averages

Using Integration to Determine Areas Between Curves

Computing Volumes of Surfaces of Revolution

Average Values

Polar coordinates and complex numbers

o Polar Coordinates o Areas in Polar Coordinates o Complex Numbers o Polar Form o Complex Exponentials and Trig Identities o Trigonometry and Complex Exponentials

Integration Techniques

Integration by parts

Trigonometric integrals

o Some Remarks on Using Complex-Valued Functions

Trigonometric substitutions

Factoring polynomials

Integration of Rational Functions Using Partial Fractions

Approximating Integrals

Regarding numerical approximation of \int_a^bf(x) dx , where f is a piecewise defined function, Sage can

compute (for plotting purposes) the piecewise linear function defined by the trapezoid rule for numerical integration based on a subdivision into N subintervals
the approximation given by the trapezoid rule,
compute (for plotting purposes) the piecewise constant function defined by the Riemann sums (left-hand, right-hand, or midpoint) in numerical integration based on a subdivision into N subintervals,
the approximation given by the Riemann sum approximation.

sage: f1 = x^2      
sage: f2 = 5-x^2
sage: f = Piecewise([[(0,1),f1],[(1,2),f2]])
sage: f.trapezoid(4)
Piecewise defined function with 4 parts, [[(0, 1/2), x/2], [(1/2, 1), 9*(x - 1/2)/2 + 1/4], [(1, 3/2), (x - 1)/2 + 5/2], [(3/2, 2), 11/4 - 7*(x - 3/2)/2]]
sage: f.riemann_sum_integral_approximation(6,mode="right")
19/6
sage: f.integral()
3
sage: n(f.integral())
3.00000000000000

Improper Integrals

o Convergence, Divergence, and Comparison

Integral_Calculus