1. Do WebAssign 7.7 #8 - #11.
Z
1
√
2.
0
Z
3.
0
π
10
dφ
θ3 + θ
5. Decide if the integral converges or diverges. If it converges, find it’s value, or give a bound on its value.
Z 1
−3/2
dx
sin(x)
0
2 − sin(φ)
dφ
φ2
4. (a) Circle one: f (t) = ln(t) is increasing, decreasing
on (0, 1].
√
(b) Insert ≥ or ≤: x
x on (0, 1].
(c) Explain
how (a) and (b) together show that
√
ln( x) ≥ ln(x) on (0, 1].
(d) Use (c) and the result from the quiz to show that
Z
1
ln
0
√ x dx converges.
6. For what values of p does the integral converge/diverge.
Z
1
2
dx
dx
x [ln(x)]p
Areas and Volumes
Understand
Understand
Apply
Apply
Apply
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Section 8.1
February 16, 2016
Name:
Math 129 - 20
Given a stack of slices, correctly identify the variable that will be used for integration.
Given a stack of slices, correctly identify the area or volume of the approximating shapes.
Express the approximate area or volume of the slices in terms of the variable of integration.
Express the approximate area (or volume) or a stack of slices in the form of a Riemann Sum.
Express the exact area (or volume) as a definite integral by taking a limit.
Idenify the correct bounds of the definite integral
In this problem, we will demonstrate how to calculate the exact
area of a circle by slicing it horizontally.
l
∆h
h
(b) The slices will be stacked in the same direction of h,
use the fact that a h2 + ( 2l )2 = 32 to express the area
of the rectangle in terms of h and ∆h.
3
The horizontal edges of each slice are
straight, but the vertical edges are curved. Explain why
the curved walls can be approximated by straight lines.
(a) Express the approximate area of the shaded slice in
terms of l and ∆h.
(c) Write a Riemann sum to approximate the total area
or the circle by stacking rectangles.
(d) Write your Riemann sum as a definite integral by taking the limit as the number of height of each slice goes
to zero (or simultaneously, as the number of slices goes
to infinity).
(Make sure the bounds on the definite integral are
correct, they should be −3 and +3.)
Quiz (Leave this space blank)