Calculus 1 Assignment 8
Alex Cowan
[email protected]
Due Thursday, December 10th at 5 pm
1. In class I defined a Riemann sum as
n
X
(xj − xj−1 )f (xj )
j=1
where the xj s are numbers which cut up a given interval [a, b]. Typically one takes the xj s to be evenly spaced,
i.e. xj = a + j b−a
n , but there are several conventions for what one should use as the height of the rectangle.
In the definition I gave, the height of the rectangle is the value of the function at the right end point of each
mini-interval [xj−1 , xj ]. This question will explore other choices.
a) Draw a picture of the rectangles associated to the sum
val [0, 1], for evenly spaced xj .
Pn
j=1 (xj
− xj−1 )f (xj ) for f (x) = x on the inter-
b) The same as a), but pick the xj s to not be evenly spaced. They still have to chop up the interval finely
as you add more and more of them (i.e. as n goes to infinity), and there’s no way to be sure from just one
picture, but pick something reasonable. Don’t pick all of them in the first half of the interval, for example.
If f is a strictly increasing function, is this way of choosing rectangles going to lead to an overestimate or an
underestimate of the actual area? What about if f is a decreasing function?
Pn
c) Draw a picture of the rectangles associated to the sum
j=1 (xj − xj−1 )f (xj−1 ) for f (x) = x on [0, 1],
with the xj s spaced however you like (evenly is fine). (The difference is the f (xj ) became an f (xj−1 )). Is this
an overestimate or an underestimate of the actual area for increasing/decreasing f ?
d) Do the same for the sum
n
X
(xj − xj−1 )f
j=1
xj + xj−1
2
Is this an overestimate or an underestimate in general? Draw pictures to substantiate your claim.
e) One slightly more sophisticated thing to do is to use trapezoids instead of rectangles, where the top is
slanted to match the function better. Draw a picture of the trapezoids corresponding to the sum
n
X
(xj − xj−1 )
j=1
f (xj ) + f (xj−1 )
2
for f (x) = 1 + x2 , and xj s evenly spaced on [0, 1].
2. Define A(x) =
Rx
0
f (t)dt. Argue that A is an antiderivative of f .
3. Suppose that f is such that f (x) ∈ [−2, 3] for all x. How big can
be?
1
R 11
5
f (x)dx be? How small can it
4. Use the chain rule to evaluate
d
dx
R x2
1
f (t)dt.
5. REvaluate the following integrals:
a) x sin(x2 )dx
R
n
b) R (logt t) dt
c) 2u sin(2u )du
Rπ
3
d) −1 log(x)ex −3 dt
R2 1
e) 1 exx2 dx
R4
f) 3 (x + 1)5 dx
R 5π
2
sin(x)3 cos(x)dx
g) −5π
2
R 3 dx
h) 2 3+t
2
R 3 dt x
i) 2 3+t2 x
6.
R4
R2
a) Suppose 0 f (x)dx = 10. What is 0 f (2x)dx?
R9
R3
b) Suppose 0 f (x)dx = 4. What is 0 xf (x2 )dx?
2