C Roettger, Spring 15
Math 165 – worksheet for ch. 5, Integration
Chapter 5 covers topics including
• Antiderivatives (OK, holdover from chapter 4, but resumed in 5.5)
• Riemann Sums, Sigma Notation (5.2)
• Simple Integrals, connection to Area (5.3)
• FTC I (derivative of accumulation function is integrand) (5.4)
• FTC II (evaluation theorem) (5.4)
• Substitution Method (5.5)
• Area between curves (5.6)
Problem 1 Solve the initial value problem y = f (t) with
y 0 = 2t + 5 cos(πt)
y(2) = 18
Problem 2 a) Express the sum S = 1/1 + 1/2 + 1/3 + · · · + 1/100 using
Sigma notation.
b) Estimate the sum by comparing it to an integral over 1/x.
Problem 3 Find the integral of the function f (t) graphed below between
x = 0 and x = 13.
Problem 4 Find the derivative of the following functions.
Z z
et
dt,
a)
f (z) =
5 2 + sin t
Z 4
1
b)
g(u) =
dt
u t(1 + cos t)
Z y
2
sin(t ) dt
c)
h(y) = sin
3
Z sin x
t3
d)
L(x) =
dt.
2
cos x t + 1
4
Problem 5 a) Find the area under the graph of x3 e−x between x = 0 and
x = 5.
4
b) Find the average value of x3 e−x between x = 0 and x = 5.
Problem 6 If f (x) is continuous on [0, a], find the integral
Z a
f (x)
M=
dx.
0 f (x) + f (a − x)
Try making the substitution u = a − x, and adding the resulting integral to
M.
Problem 7 Find the total area enclosed by the curves
y = x3 − x + 4
y = x3 − x + x2