1
1. Approximate the area under the curve y = 1+x
2 over the interval [0, 2] using a Riemann
sum approximation with n = 4 rectangles and using a lower sum estimate, that is, use the
right endpoint of each subinterval in the Riemann sum.
(12 pts.)
2. Find an upper bound on the error |ET | of a finite sum approximation of
n = 4 trapezoids. You may assume that
|ET | 
(b
a)3 M
, if |f 00 (x)|  M on [0, 2].
12n2
1
R2
1
0 1+2x
dx using
(8 pts.)
3. Find the general anti-derivative.
Z
(a)
sec(3x) tan(3x) dx
(b)
Z
(c)
Z
(d)
Z
cos(⇡x)
x
1
x
px dx,
e
x/2
(6 pts. each)
+ ⇡ 3 dx
dx
p > 0.
2
4. Find the area bounded by the line y = x
6 and the curve y =
5. Find g 0 (x) if
x2 + 5x + 6.
(16 pts.)
(8 pts.)
g(x) =
Z
x2 x
sin(3t2 ) dt.
1
3
6. Integrate.
(a)
Z
(b)
Z
(c)
Z
(8 pts. each)
p
2e
p
1
p
⇡/2
1 x
x
p
dx
3x sec2 (x2 ) dx
0
e2
1
ln(x)
dx
8x
4