Define wave function:

Psi[r_, theta_, phi_]  =  
      1/Sqrt[Pi] a0^(-3/2) Exp[-r/a0]

Part A:

Plot[Psi[r, 0, 0] /. a0 -> 1, {r, 0, 4}]

Part B:

probdr  =  
      Integrate[psi^2 r^2 Sin[theta], {phi, 0, 2  Pi}, {theta, 0, Pi}]

Part C:

Plot[4 Pi r^2 Psi[r, 0, 0]^2 /. a0 -> 1, {r, 0, 4}]

Find point where derivative is zero:

Solve[D[4 Pi r^2 Psi[r, 0, 0]^2, r]/Psi[r, 0, 0]^2 == 0, r]

Part D:  Show wave function is properly normalized---
	Integrate first over phi:

int1  =  
      Integrate[Psi[r, theta, phi]^2 r^2 Sin[theta], {phi, 0, 2 Pi}]

Then over theta:

int2  =  Integrate[int1, {theta, 0, Pi}]

Then over r:

int3  =  Integrate[int2, {r, 0, Infinity}]

Note: a0>0 ---> Integral is 1.

Part E:

prob  =  
      Integrate[Psi[r, 0, 0]^2 4 Pi r^2, {r, a0/2, 3 a0/2}]

N[prob]