The graph of the function f x( ) = 16 - x. 2 is given below. .... of the given region. y = x + sinx, 0 ⤠x ⤠Ï. 28. Find the average value of f x...
4.1-4.4 Review Sheet Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Write the following limit as a definite integral on the interval [4, 9], where ci is any point in the i th subinterval.
2. The graph of the function f (x ) 16 x 2 is given below. Which of the following definite integrals yields the area of the shaded region?
n
3c i ÊÁÁÁË 3 c 3i ˆ˜˜˜¯ x i
lim
x 0 i 1
9
a.
Ê
ˆ
Ê
ˆ
3c ÁÁÁË 3 c 3 ˜˜˜¯ dc 4 9
b.
3x ÁÁÁË 3 x 3 ˜˜˜¯ dx 4
4
c.
Ê ˆ 3x ÁÁÁ 3 x 3 ˜˜˜ dx Ë ¯
16
a.
9
Ê
ˆ
3ÁÁÁË 3 c 3 ˜˜˜¯ cdc
Ê
ˆ
Ê
ˆ
0
b.
4
e.
ˆ
0
9
d.
Ê
ÁÁÁË 16 x 2 ˜˜˜¯ dx ÁÁÁË 16 x 2 ˜˜˜¯ dx
4
All of these
4
c.
ÁÁÁË 16 x 2 ˜˜˜¯ dx 0
16
d.
16
ÊÁ ˆ ÁÁ 16 x 2 ˜˜˜ dx Ë ¯
4
e.
Ê
4
1
ˆ
ÁÁÁË 16 x 2 ˜˜˜¯ dx
Name: ________________________
ID: A
Short Answer 1. Find the general solution of the differential equation below and check the result by differentiation.
9. The height above the ground of an object thrown upward from a point s 0 feet above the ground with an initial velocity of v 0 feet per second is given by the function f(t) 16t 2 v 0 t s 0 . A balloon, rising vertically with a velocity of 24 feet per second, releases a sandbag at the instant it is 20 feet above the ground. At what velocity will it hit the ground? Round your answer to three decimal places.
dT 28x 6 dx 2. Find the indefinite integral
(8t 7)dt .
3. Find the indefinite integral
13
x 8 dx.
10. Find the sum given below. 5
4. Find the indefinite integral and check the result by differentiation.
(2i 4) i1
3z 2 12z 9 dz z4
11. Use sigma notation to write the sum 5 5 5 5 . 1 22 11 12 13
5. Find the indefinite integral 10 sins 7cos s ds .
12. Use the properties of summation and Theorem 4.2 to evaluate the sum.
6. Find the indefinite integral 2secy ÊÁË tany secy ˆ˜¯ dy .
23
(2i 5) i1
7. Solve the differential equation. 13. Use the properties of summation and Theorem 4.2 to evaluate the sum.
dy 12t 2 , y (0) 5 dt
37
(i 6)
8. Solve the differential equation.
2
i1
dP 15x 4 5, P (2) 8 dx
14. Use left endpoints and 10 rectangles to find the approximation of the area of the region between the graph of the function 4x 2 x 1 and the x-axis È ˘ over the interval ÍÍÎ 4,14 ˙˙˚ . Round your answer to the nearest integer.
2
Name: ________________________
ID: A
15. The diagram below shows upper and lower sums for the function
18. Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. 9
using 4 subintervals. Use upper and lower sums to approximate the area of the region using the 4 subintervals.
81 s 2 d s
0
19. Evaluate the integral. 8
(24s 1) d s 7
given, 8
x 3 dx 7
1695 , 4
8
x 2 dx
16. Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral.
7
169 , 3
8
x dx
4
7
2s d s
8
15 , 2
dx 1.
1
7
17. Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral.
20. Evaluate the definite integral of the algebraic function.
1
6
ÊÁË 1 u ˆ˜¯ d u
(5u 4)du
1
3
Use a graphing utility to verify your results. 1
3 2
21. Evaluate the definite integral of a function u du. 0
Use a graphing utility to verify your results.
3
Name: ________________________
ID: A
22. Evaluate the definite integral of the algebraic function. 5
0
26. Determine the area of the given region.
y 4x (1 x)
4 x 4dx
Use a graphing utility to verify your results. 23. Evaluate the definite integral of the algebraic function. 9
5
9 z 2 36 dz
Use a graphing utility to verify your results. 24. Evaluate the definite integral of the function. 27. Determine the area of the given region.
5x
(2t 5 cos t ) dt
y x sin x, 0 x
0
Use a graphing utility to verify your results.
È ˘ 25. Determine all values of x in the interval ÍÍÎ 1,3 ˙˙˚ for Ê ˆ 4 ÁÁ x 2 1 ˜˜ Ë ¯ equals its which the function f (x ) 2 x 16 average value . 3
Ê ˆ 5 ÁÁ x 2 5 ˜˜ Ë ¯ 28. Find the average value of f (x ) on the 2 x È ˘ interval ÍÍÎ 1,3 ˙˙˚ .
E PTS: 1 DIF: Easy Write a limit as a definite integral on an interval Section 4.3 E PTS: 1 DIF: Easy Write a definite integral for a bounded region Section 4.3
REF: 4.3.10 MSC: Skill REF: 4.3.17 MSC: Skill
SHORT ANSWER 1. ANS: T (x ) 4x 7 C PTS: 1 DIF: Easy REF: 4.1.5 OBJ: Calculate the general solution of a differential equation NOT: Section 4.1 2. ANS: 4t 2 7t C PTS: 1 DIF: Easy REF: 4.1.15 OBJ: Evaluate the indefinite integral of a function NOT: Section 4.1 3. ANS: 13 21 13 x C 21 PTS: 1 DIF: Easy REF: 4.1.23 OBJ: Evaluate the indefinite integral of a function NOT: Section 4.1 4. ANS: 3 6 3 2 3 C z z z PTS: 1 DIF: Medium REF: 4.1.28 OBJ: Evaluate the indefinite integral of a function NOT: Section 4.1 5. ANS: 10 cos s 7 sin s C PTS: 1 DIF: Easy REF: 4.1.35 OBJ: Evaluate the indefinite integral of a function NOT: Section 4.1
1
MSC: Skill
MSC: Skill
MSC: Skill
MSC: Skill
MSC: Skill
ID: A 6. ANS: 2secy 2tany C PTS: 1 DIF: Medium REF: 4.1.40 OBJ: Evaluate the indefinite integral of a function NOT: Section 4.1 7. ANS: y (t ) 4t 3 5
PTS: 1 DIF: Medium REF: 4.1.74b OBJ: Solve differential equations related to position/velocity/acceleration MSC: Application NOT: Section 4.1 10. ANS: 50 PTS: 1 MSC: Skill 11. ANS:
DIF: Easy NOT: Section 4.2
REF: 4.2.1
OBJ: Calculate a sum given in sigma notation
DIF: Easy NOT: Section 4.2
REF: 4.2.8
OBJ: Write a sum in sigma notation
22
1 5 j j1
PTS: 1 MSC: Skill 12. ANS: 667
PTS: 1 DIF: Medium REF: 4.2.18 OBJ: Evaluate a sum using summation properties NOT: Section 4.2 13. ANS: 10,471 PTS: 1 DIF: Medium REF: 4.2.19 OBJ: Evaluate a sum using summation properties NOT: Section 4.2
2
MSC: Skill
MSC: Skill
ID: A 14. ANS: 3125 PTS: 1 DIF: Medium REF: 4.2.29 OBJ: Approximate the area bounded by a function using rectangles MSC: Skill NOT: Section 4.2 15. ANS: lower: 1.166 ; upper: 1.666 PTS: 1 DIF: Medium REF: 4.2.42 OBJ: Estimate the area of a region using upper and lower sums MSC: Skill NOT: Section 4.2 16. ANS: 15 PTS: 1 MSC: Skill 17. ANS: 1
DIF: Easy NOT: Section 4.3
REF: 4.3.25
OBJ: Evaluate a definite integral geometrically
PTS: 1 MSC: Skill 18. ANS: 81 4
DIF: Easy NOT: Section 4.3
REF: 4.3.29
OBJ: Evaluate a definite integral geometrically
PTS: 1 MSC: Skill 19. ANS: –179
DIF: Easy NOT: Section 4.3
REF: 4.3.31
OBJ: Evaluate a definite integral geometrically
PTS: 1 DIF: Easy REF: 4.3.37 OBJ: Evaluate the definite integral of a function NOT: Section 4.3 20. ANS: 79.5 PTS: 1 DIF: Medium REF: 4.4.7 OBJ: Evaluate the definite integral of a function NOT: Section 4.4 21. ANS: 2 5 PTS: 1 DIF: Medium REF: 4.4.16 OBJ: Evaluate the definite integral of a function NOT: Section 4.4
3
MSC: Skill
MSC: Skill
MSC: Skill
ID: A 22. ANS: 98 PTS: 1 DIF: Medium REF: 4.4.23 OBJ: Evaluate the definite integral of a function NOT: Section 4.4 23. ANS: 618 PTS: 1 DIF: Difficult REF: 4.4.25 OBJ: Evaluate the definite integral of a function NOT: Section 4.4 24. ANS: 25 2 PTS: 1 DIF: Easy REF: 4.4.34 OBJ: Evaluate the definite integral of a function NOT: Section 4.4 25. ANS: x 3
MSC: Skill
MSC: Skill
MSC: Skill
PTS: 1 DIF: Easy REF: 4.4.52b OBJ: Identify the points where a function equals its average value over a given interval MSC: Skill NOT: Section 4.4 26. ANS: 2 3 PTS: 1 MSC: Application 27. ANS: 0.5 2 2
DIF: Medium NOT: Section 4.4
REF: 4.4.35
OBJ: Calculate the area bounded by a function
PTS: 1 MSC: Application 28. ANS: 40 3
DIF: Medium NOT: Section 4.4
REF: 4.4.38
OBJ: Calculate the area bounded by a function
PTS: 1 DIF: Easy REF: 4.4.52a OBJ: Calculate the average value of a function over a given interval MSC: Skill NOT: Section 4.4
