Calculus I: Section 4.3

Andy Long

Examples, p. 318:

    #3 (don’t forget about thinking of integrals as areas).
    #8
    #13 (a composition, requiring the chain rule)
    #22
    #23
    #58

#22:

Integrate[1+1/2*u^4-2/5*u^9,u]
Integrate[1+1/2*u^4-2/5*u^9,{u,0,1}]

#23:

Integrate[Sqrt[x],x]
Integrate[Sqrt[x],{x,1,9}]

#3: you didn’t need to go to all this trouble. But I thought that I would!:)

f[x_]:=
Piecewise[
{
{2,x<1}
,{2+2(x-1),x<2}
,{4-4(x-2),x<3}
,{0-(x-3),x<5}
,{-2,x<6}
,{-2+2(x-6),x<7}
}]
Plot[f[x],{x,0,7}]
g[x_]:=Integrate[f[t],{t,0,x}]
Plot[g[x],{x,0,7}]

#8: is a lot easier than you might imagine!

#11: requires a twist...

#13 (a composition, requiring the chain rule)

#58

sinc[x_]:=Sin[x]/x
Plot[sinc[x],{x,-4Pi,4Pi}]
(* Actually Sinc is built-in to Mathematica: *)
Si[x_]:=NIntegrate[Sinc[t],{t,0,x}]
Plot[Si[x],{x,-4Pi,4Pi},PlotRange-> {{-4Pi,4Pi},{-2,2}}]

FindRoot[Si[x]==1,{x,Pi/2}]
FindRoot[Sinc'[x]==0,{x,4}]