Study Guide E4
(MAT250 - 1E1)
Find the most general antiderivative.
1)
y
5 + 3
y
∫dy
2)
sin θ(cot θ + csc θ)
∫
dθ
3)
(8e5x - 7e-x)
∫ dx
4)
6
1 + y2 -
7
y
∫
dy
Use a finite approximation to estimate the area under the
graph of the given function on the stated interval as
instructed.
5)
f(x) = x
2
between x = 0 and x = 2 using an
upper sum with two rectangles of equal width.
6)
f(x) = x
2
between x = 3 and x = 7 using the
"midpoint rule" with four rectangles of equal
width.
Use a finite sum to estimate the average value of the
function on the given interval by partitioning the interval
and evaluating the function at the midpoints of the
subintervals.
7)
f(t) = 3 - cos
π
t
2
2
on [0, 4] divided into 4
subintervals
Write the sum without sigma notation and evaluate it.
8)
3
k = 1
(-1)k+1 cos 4kπ
∑
9)
3
k = 1
6k cos kπ
∑
Find the value of the specified finite sum.
10)
Given
n
k=1
ak∑= 2 and
n
k=1
bk∑= 8, find
n
k=1
ak - 2bk
∑.
Graph the function f(x) over the given interval. Partition
the interval into 4 subintervals of equal length. Then add
to your sketch the rectangles associated with the Riemann
sum
4
k=1
f(ck) Δxk
∑, using the indicated point in the kth
subinterval for ck.
11)
f(x)
=
cos x
+
3
, [0, 2
π
], right
-
hand endpoint
x
π
2π3π
22π
y
5
4
3
2
1
x
π
2π3π
22π
y
5
4
3
2
1
Find the formula and limit as requested.
12)
For the function f(x) = 9 - 3x
2
, find a formula
for the lower sum obtained by dividing the
interval [0, 1] into n equal subintervals. Then
take the limit as n→∞ to calculate the area
under the curve over [0, 1].
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