Question 1

f[x_]=((x(4-x))/2)+Sin[5π x/4];
g[x_]=(1/2)Sin[7π x/4];
Plot[{f[x],g[x]},{x,-1,5},PlotStyle->{Red,Blue}]

(∫_0^4)(f[x]-g[x])\[DifferentialD]x

Question 2

f[x_]=Sin[x];
(∫_0^π)(Sqrt[1+((f'[x])^2)])\[DifferentialD]x
N[%]

Question 3

f[x_]=2Sin[2x];
g[x_]=(x^2);
c=x/.FindRoot[f[x]==g[x],{x,1}][[1]];
Plot[{f[x],g[x]},{x,0,c},PlotStyle->{Red,Blue}]

(a) To revolve around the x-axis, we use disks.

R[x_]=f[x];r[x_]=g[x];
(∫_0^c)π(((R[x])^2)-((r[x])^2))\[DifferentialD]x

(b) To revolve around the y-axis, we use shells:

(∫_0^c)2π(R[x]-r[x])\[DifferentialD]x

Question 4

∫(1+Log[x])(Sqrt[1+(((x Log[x]))^2)])\[DifferentialD]x

u[x_]=x Log[x];
u'[x]

∫(Sqrt[1+u])\[DifferentialD]u/.u->u[x]