The Fundamental Theorem of Calculus

Formally speaking, the fundamental theorem of calculus is split into two parts.

Loosely speaking, and for purposes here, we may say that the fundamental theorem of calculus states that

\int_a^b f(x) \, dx=F(b)-F(a)

where F(x) is the antiderivative of f(x), that is to say

F

 It follows from the Fundamental Theorem of Calculus that

\int_4^1 2x^3\, dx=F(4)-F(1)=\frac{1}{2}(4)^4-\frac{1}{2}(1)^4=128-\frac{1}{4}=127\frac{3}{4}

Definite integrals are integrals that have limits. Limits appear in pairs; a number to the right of the top of the integral a number to the right of the bottom of the integral, such as in the following:.

 \int_a^b f(x)\, dx


Integrating Polynomials

Integration can be thought of as the opposite of differentiation, although the fundamental theorem of calculus isn’t quite this simple.

Differentiating a polynomial term, such as x^3, requires first multiplying down by the power then reducing the power by one. If y=x^3, \frac{dy}{dx}=3x^2. Integration is the reverse process. Given \frac{dy}{dx}=3x^2, one can find y by integrating, i.e, increasing the power by 1 then dividing by the new power: y=\frac{3x^3}{3}=x^3.

Given any y, we can write its integral as

\int y\,dx

This is an indefinite integral, definite integrals also have limits. Integrating a power of x requires adding one to the power then dividing by the new power, i.e.

\int x^n\, dx=\frac{1}{n+1}x^{n+1}+c

Notice that we have added c on the end – this is known as the integration constant. The reason this is required whenever integration is performed is because integration is the reverse process of differentiation – whenever a constant is differentiated it disappears. This constant must be brought back in when the reverse process is performed. We call it c conventionally; it represents any constant value.

Example – Integrate y=6x^2-3x^5.

\int y\, dx=\int\left(6x^2-3x^5\right)dx=\frac{6}{3}x^3-\frac{3}{6}x^6+c=2x^3-\frac{1}{2}x^6+c

Example – calculate \int_1^4 2x^3\, dx.

If f(x)=2x^3, then the antiderivative is given by

F(x)=\frac{2}{4}x^4=\frac{1}{2}x^4

Example – \int^3_0 x^2\, dx=\left[\frac{1}{3}x^3\right]^3_0=\left(\frac{1}{3}(3)^3\right)-(\frac{1}{3}(0)^3)=9-0=9.

Example – \int x^2\, dx=\frac{1}{3}x^3+c

 


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