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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a theorem that connects the two branches of calculus, differential and integral, into a single framework. We saw the computation of antiderivatives previously is the same process as integration; thus we know that differentiation and integration are inverse processes. The Fundamental Theorem of Calculus formalizes this connection. The theorem is given in two parts.

First Fundamental Theorem of Calculus

The first Fundamental Theorem of Calculus states:

If f is continuous on an interval [a,b], then the function g defined by:

g(x)=∫xaf(t) dta≤x≤b

is continuous on the interval [a,b] and differentiable on (a,b) and g′(x)=f(x).

Second Fundamental Theorem of Calculus

The second Fundamental Theorem of Calculus states:

If f is continuous on the interval [a,b] then:

∫baf(x) dx=F(b)−F(a)

Where F is any antiderivative of f

Examples

In [2]:
from sympy import symbols, limit, diff, sin, cos, log, tan, sqrt, init_printing, plot, integrate
from mpmath import ln, e, pi

init_printing()
x = symbols('x')
y = symbols('y')

Example 1: Evaluate the integral: ∫2−1(x3−2x) dx

Applying the second part of the Fundamental Theorem of Calculus, we take the antiderivative of the function and evaluate the integral.

∫2−1(x3−2x) dx=14x4−x2|2−1=14(−1)4−(−1)2−14(2)4−(2)2=34

We can verify our answer using SymPy's integrate() function.

In [3]:
integrate(x ** 3 - 2 * x, (x, -1, 2))
Out[3]:
34

Example 2: Evaluate ∫41(5−2x+3x2) dx

As in the previous example, we take advantage of the second part of the Fundamental Theorem of Calculus:

∫41(5−2x+3x2) dx=5x−x2+x3|41=5(4)−(4)2+(4)3−5(1)−(1)2+(1)3=63
In [5]:
integrate(5 - 2 * x + 3 * x ** 2, (x, 1, 4))
Out[5]:
63

Example 3: Compute the integral ∫10x45 dx

∫10x45 dx=59x95|10=59(1)95−59(0)95=59
In [21]:
integrate(x ** (4/5), (x, 0, 1)) # Returned result will be in decimal form.
Out[21]:
0.555555555555556

Example 4: Determine the integral ∫213x4 dx

Rewriting the integral as ∫213x−4 dx:

∫213x−4 dx=−x−3=−1x3|21=−1(2)3+1(1)3=−18+1=78
In [11]:
integrate(3 / x ** 4, (x, 1, 2))
Out[11]:
78

Example 5: Compute the integral ∫20x(2+x5) dx

Start by factoring:

∫202x+x6 dx=x2+17x7|20=(2)2+1727−(0)2+17(0)7=4+1287=287+1287=1567
In [20]:
integrate(x * (2 + x ** 5), (x, 0, 2))
Out[20]:
1567

References

Fundamental theorem of calculus. (2017, December 2). In Wikipedia, The Free Encyclopedia. From https://en.wikipedia.org/w/index.php?title=Fundamental_theorem_of_calculus&oldid=813270221

Stewart, J. (2007). Essential calculus: Early transcendentals. Belmont, CA: Thomson Higher Education.

Weisstein, Eric W. "Fundamental Theorems of Calculus." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html

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