193
Worksheet 4.9.
Antiderivatives
- + 36* dz
Evaluate the indefinite integral.
1. f (5x3 - x~* - x
7. Solve the differential equation -p = Sx3 + 3s2 - 3 with initial condition j/(l) = 1.
2.
3.
,
dx
8. Given that /"(*) = x3 - 2x + 1, /'(O) = 1, and /(O) = 0, first find /' and then find./.
5. /'
213
Worksheet 5.1.
Approximating and Computing Area
3. Evaluate the following sums (see Equations (3)-(5)). (optional)
20
1. Calculate the area of the shaded rectangles in the figure. Which approximation to the
area under the curve is this?
4-x
- 3 - 2 - 1
1
2
3
-2
4. Evaluate lira Y^
-
*—*
• ,
^—.
(optional)
3
2. Estimate Re and L$ for the function shown in the graph.
5. Use Equations (3)-(5) to find a formula for RN for f(x) = 3x2 - x + 4 over the interval
[0,1]. (optional)
0.5
1
1.5
2
2.5
3
1 N I
( ' \2
6. Evaluate lira — ^ < / l - f — I by interpreting the limit as an area, (optional)
j=\ V
\ /
219
)
220
Worksheet 5.2.
The Definite Integral
{
.j. _ i
» _ 2
ifO^rE^
TS ~
<
to
evaluate the integrals.
5
1. Calculate /f (2z + 1) dx in two ways: as a limit wlim RN and using geometry.
Jt
-»~
-2
r3
a. / g(x) dx
Jo
j - X / l -(x-1) 2 i f O < o : < 2
2. The graph of /(a;) = <1
consists of two semicircles:
5
b. /f g(x) dx
Jo
4. Use the basic properties of the integral and Equations (4)-(6) to calculate the following
integrals.
Evaluate the following integrals.
/•»
a. / /(z) dx
b. /*/(*)<*«
Jo
f*
I \2x-4\dx
Ji
'
'
Worksheet 4.9.
Antiderivatives
)
* dx
3. / ( - +3e*j
C.
Evaluate the indefinite integral.
1. /(5x3 - a-2 - x3'*)dx
7. Solve the differential equation y^ = 8.T3 + 3.T2 - 3 with initial condition j/(l) = 1.
' + 2x - 3
8. Given that /"(a;) = .r3 - 2.i- + 1, /'(O) = 1, and /(O) = 0, first fiiid /' and then find /.
1. / sin9:c
5. / 18sin(3
f =
O^ c
Worksheet 5.1.
Approximating and Computing Area
1. Calculate the area of the shaded rectangles in the figure. Which approximation to the
area under the curve is this?
&X~
2. Estimate 7?6 and L6 for the function shown in the graph
j
0.5
1
1.5
2
2.5
3
-5:3
>
,
~3
I
21!)
Worksheet 5.2.
The Definite Integral
3. Let n(x) mi*
C
,
I o — Z.')<
jgrals.
f 'r!. Use the graph of g(x) to evaluate the integn
11 o ^ X ^ 0
r
1. Calculate / (2:r + l)</j; in two ways: as a limit lira RN and using geometry.
J-2
N~too
4
5
6
-2
a. I* <,(x)dx
2. The graph of/(.T) = S
-V/l - (.r-1) 2 if 0 < .1 < 2
,
rr
consists of two semicircles:
:
- (.r - 4) 2 if 2 < x < 6
2.TT
-'
s -feO*-/ •
/
4. Use the basic properties of the integral and Equations (4)- (G) to. calculate the following
integrals.
'
/^
Evaluate the following integrals. '
jf/w- ^
'^rA
•*• J^^-J
/>
= - - - * +1
= -3- -V --
/+/ ^