Properties of Definite Integral

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Now let's see what properties integral has.

Property 1. It follows from the definition of integral that int_a^af(x)dx=0.

Property 2. Inverting bounds of integration: int_a^bf(x)dx=-int_b^af(x)dx.

Property 3. If f is an even function then int_(-a)^af(x)dx=2int_0^af(x)dx.

In terms of areas this means following: if f is symmetric about x-axis then area from -a to 0 equals area from 0 to a.

Property 4. If f is an odd function then int_(-a)^af(x)dx=0.

In terms of areas this means following: if f is symmteric about origin then area from -a to 0 equals minus area from 0 to a. This means that net area is 0.

Adjacency of Integral: int_a^bf(x)dx=int_a^c+int_c^bf(x)dx for any displacement of points a, b and c.

Example 1. It is known that int_0^8f(x)dx=15 and int_5^8f(x)dx=11. Find int_0^5f(x)dx.

Since int_0^8f(x)dx=int_0^5 f(x)dx+int_5^8 f(x)dx then 15=int_0^5 f(x)dx+11.

So, int_0^5 f(x)dx=4.

Constant Multiple Rule for Integrals: Constant can be factored out of integral sign: if  is a constant then int_a^bkf(x)dx=kint_a^bf(x)dx.

Sum (Difference) Rule for Integrals: int_a^b(f(x)+-g(x))dx=int_a^bf(x)dx+-int_a^b g(x)dx. Note that this rule is true for any number of functions.

Example 2. Use properties of integrals to evaluate int_(-1)^1(3+2sqrt(1-x^2))dx

Using properties of integral we have thatint_(-1)^1(3+2sqrt(1-x^2))dx=int_(-1)^1 3dx+int_(-1)^1 2sqrt(1-x^2)dx=int_(-1)^1 3dx+2int_(-1)^1 sqrt(1-x^2)dx=

=3*(1-(-1))+2*pi/2=pi+6

because int_(-1)^1sqrt(1-x^2)dx=pi/2 is area of semicircle with radius 1 and int_(-1)^1 3dx is area under line y=3 on interval [-1,1] (clearly it is area of rectangle with sides 3 and 1-(-1)=2).

Comparison Properties of the Integral
Comparison Property 1. If f(x)>=0 for a<=x<=b then int_a^b f(x)dx>=0 (area under the graph of non-negative function should be non-negative).

Comparison Property 2. If f(x)>0 for a<=x<=b then int_a^b f(x)dx>0 (area under the graph of positive function should be positive).

Comparison Property 3.If f(x)>=g(x) for a<=x<=b then int_a^b f(x)dx>=int_a^b g(x)dx (area under the graph of f should be bigger than area under the graph of g because f takes bigger values on all interval). This property follows from property 1 because f-g>=0.

Comparison Property 4. |int_a^bf(x)dx|<=int_a^b |f(x)|dx. Geometrically this means that absolute value of net area is not greater then sum of absolute values of areas. This inequality is very similar to triangle inequality |a+b|<=|a|+|b|.

Comparison Property 5. If m<=f(x)<=M for a<=x<=b then m(b-a)<=int_a^b f(x)dx<=M(b-a). It says that if m is an absolute minimum and M is absolute maximum of f on interval [a,b] then area under the graph of f is greater then area of rectangle with height m and lesser then area of rectangle with height M. Looking on figure we can say that area under curve is less than area of rectangle ACDF and greater than area of rectangle ABEF. Note that this property gives very rough estimate.

Example 3. Use Comparison Property 5 to estimate int_1^2 sqrt(x^3+1)dx.

f(x)=sqrt(x^3+1) and f'(x)=1/(2sqrt(x^3+1))*(x^3+1)'=3/2 (x^2)/(sqrt(x^3+1)).

As can be seen f'(x)>0 for all x on interval [1,2], so f(x) is increasing on interval [1,2].

This means that on interval [1,2] absolute minimum is m=f(1)=sqrt(1^3+1)=sqrt(2) and absolute maximum is M=f(2)=sqrt(2^3+1)=3.

Therefore, sqrt(2)*(2-1)<=int_1^2 sqrt(x^3+1)dx<=3*(2-1) or 1.414<=int_1^2 sqrt(x^3+1)dx<=3.

Mean Value Theorem for Integrals. If m<=f(x)<=M for a<=x<=b then there exists such number n that m<=n<=M and int_a^b f(x)dx=n(b-a).

Since f(x) is continuous then equivalently there exists number c such that a<=c<=b and int_a^b f(x)dx=f(c)(b-a).

Geometrically mean value theorem for positive functions means following: area under curve y=f(x) equals area of rectangle whose height lies somewhere between minimum and maximum values of f on interval [a,b]. In other words area under f(x) equals area of rectangle ABCD (sum of yellow areas equals green area).

Note that there can be more than one point with above property. For example, on the figure there is another point d such that n=f(d).

Generalized Mean Value Theorem for Integrals. If m<=f(x)<=M for a<=x<=b and g(x) is either positive or negative on interval [a,b] (in other words g(x) doesn't change sign) then exists number n such that m<=n<=M and int_a^b f(x)g(x)dx=n int_a^bg(x)dx.

If f is continuous then we can equivalently write that there exists number c such that a<=c<=b and int_a^b f(x)g(x)dx=f(c)int_a^b f(x)dx.