Mr. Orchard’s Math 142 WIR
1. Use the table below to estimate
x
0 1
2 3 4 5
f (x) 8 −2 −3 4 6 5
6.4-6.6
R5
0
Week 12
f (x)dx with n = 5 equal subintervals:
(a) Left hand sum
(b) Right hand sum
2. Use a left sum and a right
R 5 sum with n = 5 rectangles of equal width to find an upper
and a lower bound for 0 (4 − x2 )dx.
(a) Upper bound
(b) Lower bound
Mr. Orchard’s Math 142 WIR
6.4-6.6
Week 12
3. Below is the graph of g(x). Use it to find the following integrals.
4
3
2
1
0
-1
-2
-3
-4
-4
(a)
R0
(b)
R4
(c)
R4
−4
0
g(x)dx
g(x)dx
−4
g(x)dx
-3
-2
-1
0
1
2
3
4
Mr. Orchard’s Math 142 WIR
6.4-6.6
Week 12
4. On the interval [−3, 4], we know −2 ≤ f (x) ≤ 1. Use this to show −14 ≤
5. (a) Use a left hand sum with n = 4 equal rectangles to estimate
(b) Find
R2
(c) Find
R2
−2
−2
R2
−2
R4
−3
f (x)dx ≤ 7.
(x + 1)dx.
(x + 1)dx by finding the graph of an appropriate geometric region.
(x + 1)dx using the Fundamental Theorem of Calculus.
Mr. Orchard’s Math 142 WIR
6.4-6.6
6. Evaluate the following integrals exactly:
R 64 √
(a) 25 9 xdx
(b)
R5
(c)
R0
(d)
R e2
(e)
R2
1
dx
1 2x
−3
e
2
xex dx
(ln x)3
dx
x
f (x)dx where f (x) =
−3
x2 x < 0
3x x ≥ 0
Week 12
Mr. Orchard’s Math 142 WIR
6.4-6.6
Week 12
7. The rate of sales of an item is given by R0 (t) = −3t2 + 36t where t is the number of
weeks after an advertising campaign has begun and R0 (t) is measured in thousands of
dollars per week. Find the amount of sales, in thousands of dollars, for the third week?
8. If f (1) = 15, f 0 is continuous, and
R4
1
f 0 (x)dx = 17, what is the value of f (4)?
9. Use the graph of f 0 (x) (given below) and that f (3) = 5 to determine f (−6).
6
4
2
0
-2
-4
-6
-6
-4
-2
0
2
4
6
Mr. Orchard’s Math 142 WIR
6.4-6.6
Week 12
10. Find the average value of f (x) = 2e2x on the interval [3, 7].
11. The average value of f (x) = 14x + 3x2 on the interval [0, B] is 18. If B > 0, what is the
value of B?
12. Find the area between y = 4 − x2 and y = x + 1 on the interval [−1, 1].
Mr. Orchard’s Math 142 WIR
6.4-6.6
13. Find the area bounded by the curves y = ex , y = e3x and x = −1.
14. Find the area bounded by the curves y = x3 and y = x.
Week 12