Assignment 2 - Math 105 - Section 211
Hand in your solutions on Wednesday, February 27, at the beginning of class.
1. (a) Estimate the net area between the graph of f (x) = 1 − x2 and the x-axis from x = −1 to
x = 2 using a right Riemann sum with n = 3. Illustrate with a sketch.
(b) Repeat part (a) using a left Riemann sum.
(c) Repeat part (a) using a midpoint Riemann sum.
n
X
2
2k 10
2. Determine a region whose area is equal to lim
5+
.
n→∞
n
n
k=1
3. Express lim
n
X
n→∞
Z
i=1
25 i4
as a definite integral on the interval [0, 2].
n5
2
x3 dx as a limit of Riemann sums.
4. Evaluate
−1
5. (a) Sketch the region bounded by the graph of y = 1 − 12 |x| and the x-axis from x = −2 to x = 4.
Z 4
(b) Evaluate
(1 − 21 |x|) dx by interpreting it in terms of areas.
−2
Z
2
6. Evaluate
√
| 3 x − 1| dx.
−7
Z
b
7. Find the values of b at which
(9x − x3 )ecos x dx has a local maximum.
−1
Z
8. Evaluate
x3 cos(x2 + 1)dx. Check your work by differentiation.
9. Evaluate the following integrals.
Z
dx
√
(a)
x 9x2 − 36
Z 1
√
(b)
( 4 u + 1)2 du
Z
x3 cos x dx
Z
x cos3 x dx
(i)
(j)
0
Z
(c)
Z
4x4 + 16x3 + 21x2 + 13x + 2
dx
4x3 + 4x2 + x
(k)
sin(ln x) dx
(l)
Z
Z
(d)
dx
2 x+3+x
√
π/4
p
1 + cos(4x) dx
−π/4
Z
dx
sin x − 1
(e)
Z
(cot2 x + cot4 x) dx
(f)
Z
(g)
Z
(h)
0
2x2 − 8x + 4
√
dx
4x − x2
1
ez + 1
dz
ez + z
Z
√
x ln(8 x) dx
Z
4x2 − 3x + 2
dx
4x2 − 4x + 3
(m)
(n)
π/3
sin t sec2 t
dt
1 + sec t
0
Z
p
(p) x7 1 + x4 dx
Z
(o)