Learning Activity for MATH 115
Applied Math for Business Fall 2013
NAME:
Section:
Answer each of the questions below to the best of your ability. Be sure to answer all word problems
using full sentences that include units for the problem.
Theory
The Fundamental Theorem of Integral Calculus
If a continuous function f has an anti-derivative F over [a, b], then
Z
b
f (x) dx = F (b) − F (a)
a
Note this integral represents the net total area (the difference between the sum of the areas above
the x−axis and the sum of the areas below the x-axis.
Questions
For each of the following determine the integral.
Z
8
x4 dx
1.
4
8
Z
4
Z
1 5 8 1 5 1 5 31744
x dx = x = (8) − (4) =
= 6348.8
5 4 5
5
5
4
3
2.
5y + 3 dy
−2
Z
3
5y + 3 dy =
−2
Z
3.
6
15
4
dx
8x
Z
6
Z
4.
3
5 2
55
y + 3y =
≈ 27.5
2
2
−2
15
4
dx =
8x
15
1
1
1
ln(x) = − log (6) + log (15) ≈ 0.45815
2
2
2
6
11 √
x dx
4
Z
4
11 √
2 3 11 22 √
16
x dx =
x2 =
11 −
≈ 18.98858
3
3
3
4
Z
5.
4√
x dx
11
Z
4√
11
Z
6.
e
e3
2 3 4
16
22 √
x dx =
x2 = −
11 +
≈ −18.98858
3
3
3
11
5
dx
7x
e3
Z
e
5
dx =
7x
e3
5
10
ln(x) =
≈ 1.42857
7
7
e
7. Find the area bounded between the x-axis and the function f (x) = x4 between x = 2 and
x = 5.
Z 5
1 5 5 3093
4
x dx =
x =
≈ 618.6
5
5
2
2
8. A company estimates that its sales will grow continuously at a rate given by the function
S 0 (t) = 13et
where S 0 (t) is the rate at which sales are increasing, in dollars per day, on day t.
• Find the accumulated sales for the first 3 days.
Z 3
3
t
t 13e dt = 13e = 13 e3 − 13 ≈ 248.11198
0
0
The total sales in the first 3 days is $248.11.
• Find the sales from the start of day 4 through the start of day 6.
Z 5
5
t
t 13e dt = 13e = 13 e3 − 13 ≈ 248.11198
3
3
The total sales here is is $1668.26.
9. Pure Water Enterprises finds that the marginal profit, in dollars, from drilling a well that is
x feet deep is given by
√
P 0 (x) = 5 x.
Find the profit when a well 330 feet deep is drilled.
Here we find the profit function by integration, assuming that the profit for a well drilled 0
feet deep is nothing (P (0) = 0). Thus,
Z
P (x) =
0
P (x) dx =
Z
x1/5 dx =
P (0) = 0 =⇒ C = 0
Then the profit for a well drilled 330 feet deep is $ 877.07.
5x6/5
+C
6