Formal Definition of the Definite Integral

Key Questions

How do you use the definition of the definite integral?

The definite integral can be used for a number of different purposes.

A common use is finding the area underneath the line on a graph; in this case, the definite integral is taken between the leftmost and rightmost points of the area in question.

Another use - and my personal favourite - is finding the volume, surface area, or similar attributes of solids of revolution. Often, one needs to use the definite integral twice, or three times, in one problem.

CJ · · Sep 22 2014
What is the Formal Definition of the Definite Integral of the function #y=f(x)# over the interval #[a,b]# ?

#int_a^bf(x)dx=lim_{n to infty}sum_{i=1}^n f(a+iDelta x)Deltax#,
where #Delta x ={b-a}/n#

Wataru · · Sep 8 2014

Questions

What is the Formal Definition of the Definite Integral of the function #y=f(x)# over the interval #[a,b]# ?
How do you use the definition of the definite integral?
What is the integral of dy/dx?
What is an improper integral?
How do you calculate the double integral of #(xcos(x+y))dr# where r is the region: 0 less than or equal to x less then or equal to #(2pi)/6#, 0 less than or equal to y less than or equal to #(2pi)/4#?
How do you apply the evaluation theorem to evaluate the integral #3t dt# over the interval [0,3]?
What is the difference between an antiderivative and an integral?
How do you integrate #3x^2-5x+9# from 0 to 7?
Question #f27d5
How do you evaluate the definite integral #int sqrtt ln(t)dt# from 2 to 1?
How do you evaluate the definite integral #int 2(pi)x(cos^(-1)(x))dx# from 0 to 1?
How do you evaluate the definite integral #int -8/3*2^xdx# from -2 to 9?
How do you evaluate the definite integral #int (1 + ln x)^2 / x# from 1 to e?
How do you evaluate the definite integral #int (x+6)^2dx# from [0,4]?
How do you evaluate the definite integral #int ((6x^2+8)/(sqrt(x)))dx# from [3,9]?
How do you evaluate the definite integral #int sin^(5)x * cos^(20)x dx# from [0,pi/2]?
How do you evaluate the definite integral #int dx/(x(1+ln(x)))# from #[1,e^6]#?
How do you evaluate the definite integral #int (x^45)(cos(x^46)) dx# from #[0,(pi)^(1/46)]#?
How do you evaluate the definite integral #int 4/sin^2(z) dz# from #[pi/6, pi/2]#?
How do you evaluate the definite integral #int (6x^2 - 4e^(2x))dx# from #[0,2]#?
How do you evaluate the definite integral #int 15x^2(1+x^3)^4# from #[-1,1]#?
How do you evaluate the definite integral #int e^(sin(x))*cosx dx# from #[0,pi]#?
How do you evaluate the definite integral #int (csc(6theta)-cot(6theta))^2 theta# from #[5,6]#?
How do you evaluate the definite integral #int((sqrtx + 1)/(4sqrtx))^2 dx # from #[3,9]#?
How do you evaluate the definite integral #intsec^2(t/4) # from #[0, pi]#?
How do you evaluate the definite integral #int dx / ( x(sqrt(ln(x)))# from #[e, e^81]#?
How do you evaluate the definite integral #int sqrt[2+3t^4) dt# from #[4,7]#?
How do you evaluate the definite integral #int (2)/(2x+5) * dx# from #[1, e]#?
How do you evaluate the definite integral #int (dx/(xsqrt(lnx)))# from #[1, e^6]#?
How do you evaluate the definite integral #int|x^3+64| dx# from #[-8,0]#?
How do you evaluate the definite integral #int x^(3) dx # from #[b, 2b]#?
How do you evaluate the definite integral #intte^(-t)dt# from #[0,6]#?
How do you evaluate the definite integral #int tan^2(3x)dx# from #[-pi/10, pi/11]#?
How do you evaluate the definite integral #int t sqrt( t^2+1dt)# bounded by #[0,sqrt7]#?
How do you evaluate the definite integral #int (4x)/sqrt(x^2+1) dx# from # [-sqrt3, sqrt3]#?
How do you evaluate the definite integral #int (x^2 + 1)dx# from # [1,2]#?
How do you evaluate the definite integral #int (dx/(xsqrtlnx))# from # [1,e^3]#?
How do you evaluate the definite integral #int abs(x-1)dx# from # [0,5]#?
How do you find the definite integral for: #(2-x^2)dx# for the intervals [2,-1]?
How do you find the definite integral for: #root3(x^2)# for the intervals [-1,1]?
How do you find the definite integral for: #(1) / (sqrt(1 + x))# for the intervals [0,3]?
How do you find the definite integral for: #(x^2(x^3 + 1)^3)# for the intervals [0,1]?
How do you find the definite integral for: # x^2 + x + 4# for the intervals [0,2]?
How do you find the definite integral for: #xcos(5x^2) dx# for the intervals #[0,4sqrtpi]#?
How do you find the definite integral for: #tln(t) dt# for the intervals #[1, 10]#?
How do you find the definite integral for: #(x^43) (e^(-x^(44)) dx)# for the intervals #[0, 1]#?
How do you find the definite integral for: #xdx# for the intervals #[12, 14]#?
How do you find the definite integral for: #2(pi)x(cos^(-1)(x))dx# for the intervals #[0, 1]#?
How do you find the definite integral for: #(9-x^2)^(1/2)# for the intervals #[-3, 3]#?
How do you find the definite integral for: #(3x^2+2x+3) dx# for the intervals #[0, 6]#?
How do you find the definite integral for: #(x+1)^(2)dx# for the intervals #[-7, 0]#?
How do you find the definite integral for: #(1/x^(2))dx# for the intervals #[5, 6]#?
How do you find the definite integral for: #((2x^(2) +3)dx# for the intervals #[4, 6]#?
How do you find the definite integral for: #((x^(6))dx# for the intervals #[b, 2b]#?
How do you find the definite integral for: #(6x+3)dx# for the intervals #[3, 9]#?
How do you find the definite integral for: #[(6x^2+2) / sqrt(x)] dx # for the intervals #[1, 5]#?
How do you find the definite integral for: #10sin(x)dx# for the intervals #[0, pi]#?
How do you find the definite integral for: #e^sin(x) * cos(x) dx# for the intervals #[0, pi/4]#?
How do you find the definite integral for: # dx / (x sqrt(lnx))# for the intervals #[1, e^8]#?
How do you find the definite integral for: # e^(5x) dx# for the intervals #[0, 1]#?
How do you find the definite integral for: #2 / (4+x^2) dx# for the intervals #[0, 2]#?
How do you find the definite integral for: #int_(pi/4)^(pi/3)sin(2p) / (1+cos^2(p)) dp#?
How do you find the definite integral for: #(sqrt(b^2-a^2)) da# for the intervals #[0, b]#?
How do you find the definite integral for: #(2+3x) / (4+x2) dx# for the intervals #[0, 1]#?
How do you find the definite integral for: #sqrt(4+3(t^4))dt # for the intervals #[1, 4]#?
How do you find the definite integral for: #sin^(2) (5x) * cos^(2) (5x) dx # for the intervals #[8, 11]#?
How do you find the definite integral #int_(pi/2)^((5pi)/2) x^2 cos (1/5x)dx# ?
How do you find the definite integral for: #x dx# for the intervals #[2, 4]#?
How do you find the definite integral for: # 1/x^2# for the intervals #[0, 1]#?
How do you find the definite integral for: #x^2 dx# for the intervals #[2, 5]#?
How do you find the definite integral for: #dx /(a cos^2x + b sin^2 x)^2# for the intervals #[0, pi]#?
How do you find the definite integral: #int_0^4 4 sqrtx dx#?
How do you find the definite integral for: #((2x)/(1+x^2)^(1/3))dx# for the intervals #[0, sqrt7]#?
How do you find the definite integral for: #cos(x)# for the intervals #[-pi, pi]#?
How do you find the definite integral for: #xe^(x^2 + 2) # for the intervals #[0, 2]#?
How do you find the definite integral for: #(cos(sqrt(x)))/(sqrt(x))# for the intervals #[1, 4]#?
How do you find the definite integral of #x/sqrt(2x+1)# from #[0,4]#?
How do you find the definite integral of #(x^3+x^4(tanx))dx# from #[-pi/4, pi/4]#?
How do you find the definite integral of #3x^2 sqrt(x^3+1)dx# from #[-1, 1]#?
How do you find the definite integral of #sqrt(y+1)dy# from #[0,3]#?
How do you find the definite integral of #(1/x^2)*sin[(x-1)/x] dx# from #[1,2]#?
How do you find the definite integral of #tan x dx# from #[pi/4, pi]#?
How do you find the definite integral of #2x* (x-3)^5 dx# from #[-1, 2]#?
How do you find the definite integral of #x^2dx / (x^3 + 9)# from #[-1, 1]#?
How do you find the definite integral of #5xcos(x^2)dx# from #[0, sqrtpi]#?
How do you find the definite integral of #(dx)/ (sqrtxsin(sqrtx)# from #[pi^2, 4pi^2]#?
How do you find the definite integral of #t^3(1 + t^4)^3 dt# from #[-1, 1]#?
How do you find the definite integral of #dx/ (x+2)# from #[0, 7]#?
How do you find the definite integral of # (xdx) / ((x^2 + 1)ln(x^2 + 1))# from #[1,2]#?
How do you find the definite integral of #(3x^3 +7) dx # from #[1, 5]#?
How do you find the definite integral of #2 / (4+x^2) dx# from #[0, 2]#?
How do you find the definite integral of #cot x csc^5 x dx# from #[pi/6, pi/2]#?
How do you find the definite integral of #(dx)/(xsqrt(ln(x))# from #[e^4, e^16]#?
How do you find the definite integral of #(x^2)(sqrt5x+6)dx# from #[0, 1]#?
How do you find the definite integral of #t^7cos(t^4) # from #[pi^(1/4), (pi/2)^(1/4)]#?
How do you find the definite integral of #(sqrt(2x+1))dx# from #[1, 4]#?
How do you find the definite integral of #2/(sqrt(x) *e^-sqrt(x))# from #[1, 4]#?
How do you find the definite integral of #sqrt(y+1)*dy# from #[-1, 0]#?
How do you find the definite integral of #(e^sin(x))cos(x) dx# from #[0, pi]#?
How do you find the definite integral of #x * e^ (-x^2) dx# from #[1, sqrt2]#?
How do you find the definite integral of #(lnx^3)/x# from #[1, e]#?
How do you find the definite integral of #xsqrt(x - 40)dx# from #[40, 41]#?
How do you find the definite integral of #e^(x^2) dx# from #[2, 3]#?
How do you find the definite integral of # [x / (sqrt(x+8))] dx# from #[0, 6]#?
How do you find the definite integral of #(x dx) / [(x^2 +1)(ln (x^2 +1))]# from #[1, 2]#?
How do you find the definite integral of #x*sqrt(1-x^4) dx# from #[0, 1]#?
How do you find the integral of #1/(x^(1/2))sin(x^(1/2))#?
How do you find the integral of #x^2(sec(x^3))^2#?
How do you find the definite integral of #x^2 * sqrt (x^3 + 1) * dx# in the interval #[1,2]#?
How do you find the definite integral of #(x^3)/sqrt(16 - x^2)# in the interval #[0, 2sqrt3]#?
How do you find the definite integral of #cos^4 x sin x dx# in the interval #[0, pi/3]#?
How do you find the definite integral of #xe^(x^2 + 2) # in the interval #[0, 2]#?
How do you find the Improper integral #e^[(-x^2)/2]# from #x=-∞# to #x=∞#?
How do you find the Improper integral #int (x^2)e^[(-x^2)/2] dx # from x=-∞ to x=∞?
How do you find the antiderivative of #f(x)=x^6#?
How do you find the antiderivative of #f(x)=4x^2-6x+7#?
How do you find the antiderivative of #f(x)=12x^2-6x+1#?
How do you find the antiderivative of #f(x)=8x^3+5x^2-9x+3#?
How do you find the antiderivative of #f(x)=1/4x^4-2/3x^2+4#?
How do you find the antiderivative of #f(x)=(2x+3)(3x-7)#?
How do you find the antiderivative of #f(x)=(2x^2-5x-3)/(x-3)#?
How do you find f given #f''(x)=-2+36x-12x^2# and f(0)=2, f'(0)=1?
How do you find the antiderivative of #f(x)=x^5#?
How do you find the antiderivative of #f(x)=2x^2-8x+2#?
How do you find the antiderivative of #f(x)=1/5x^4-3/4x-1#?
How do you find the antiderivative of #f(x)=(x^3-2x^2+x)/x#?
Question #a9af4
Express the integral as a limit of Riemann sums?
Given information about the fill and empty rate, how do I find the volume after t minutes?
Why do the limits of integration need to be changed if we perform a substitution?

Introduction to Integration

View all chapters

Sigma Notation
Integration: the Area Problem
Formal Definition of the Definite Integral
Definite and indefinite integrals
Integrals of Polynomial functions
Determining Basic Rates of Change Using Integrals
Integrals of Trigonometric Functions
Integrals of Exponential Functions
Integrals of Rational Functions
The Fundamental Theorem of Calculus
Basic Properties of Definite Integrals

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Formal Definition of the Definite Integral

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Introduction to Integration