http://www.ck12.org Chapter 4. Integration
- By division, we have
f(u)≤F(xx)−−Fc(c)≤f(v).- Whenxis close toc,then bothf(u)andf(v)are close tof(c)by the continuity off
- Hence limx→c+F(xx)−−Fc(c)=f(c).Similarly, ifx<c,then limx→c−F(x)x−−Fc(c)=f(c).Hence, limx→cF(xx)−−Fc(c)=
f(c). - By the definition of the derivative, we have that
F′(c) =limx→cF(xx)−−Fc(c)=f(c)for everyc∈[a,b].Thus,Fis an antiderivative offon[a,b].
Multimedia Link
For a video presentation of the Fundamental Theorem of Calculus(15.0), see Fundamental Theorem of Calculus,
Part 1 (9:26).
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URL: http://www.ck12.org/flx/render/embeddedobject/565Review Questions
In problems #1–4, sketch the graph of the functionf(x)in the interval[a,b].Then use the Fundamental Theorem of
Calculus to find the area of the region bounded by the graph and thex−axis.
- f(x) = 2 x+ 3 ,[ 0 , 4 ]
- f(x) =ex,[ 0 , 2 ]
- f(x) =x^2 +x,[ 1 , 3 ]
- f(x) =x^2 −x,[ 0 , 2 ]
(Hint: Examine the graph of the function and divide the interval accordingly.)
In problems #5–7 use antiderivatives to compute the definite integral.
5.∫−+ 11 |x|dx
6.∫ 03 |x^3 − 2 |dx(Hint: Examine the graph of the function and divide the interval accordingly.)
7.∫−+ 24 [|x− 1 |+|x+ 1 |]dx