Right Riemann Sums

1. Let f(x) = x
   2 + 1, P = f2; 1; 2; 5g. Find the left and right Riemann Sums for f over the partition
   P. What is kPk ?
2. Find Z 1
   0
   (x
   2 + x)dx by taking the limit of Right Riemann Sums, over a uniform partition of [0; 1].
3. Find limn!1
   
   2n
   Xn
   k=1
   cos
   k
   2n
4. By interpeting the integral in terms of area Önd Z
   p
   2

p
2
p
2 x
2dx

5. For the following, compute dy
   dx :
   (a) y =
   Z x
   0
   1
   t
   3 + 1
   dt
   (b) y =
   Z x
   2+1
   0
   p
   costdt
   (c) y =
   Z xex
   1
   sin
   
   d
   (d) y =
   Z y sin x
   0
   e
   t
   2
   dt
6. Compute the given integral.
   (a) Z 2
   1
   (3t
   2 4t + 7) dt
   (b) Z
   tan4 x sec2 xdx
   (c) Z e
   1
   (x
   1
   x
   ) dx
   (d) Z
   x
   p
   x
   2 + 1 dx
   (e) Z
   x
   p
   x 1 dx
   (f) Z e
   2
   e
   ln x
   x
   dx
   (g) Z
   x
   2
   p3
   x
   3 + 8
   d