alic
amindji
4
1. 21.5 Lesson 21 Problem Set
(1) Sketch the graph of f (x) = 2x3 on the interval [ 2, 3]. Shade the signed area represented
Z 2
by
f (x) dx, indicating the regions counted as positive area and negative area. Without
1
actually evaluating the definite integral, decide if the value of the integral is positive or
negative. Briefly explain your answer.
(2) Suppose
Z
1
f (x) dx = 3 and
0
calculate
Z 3
(a)
f (x) dx
1
Z 1
(b)
f (x) dx
3
Z 2
(c)
f (x) dx
2
Z 3
(d)
2f (x) dx
Z
3
f (x) dx = 2. Using the properties of definite integrals,
0
0
R2
(3) Draw the graph of f (x) = 2x + 1. Evaluate 2 f (x) dx by thinking of the integral as the
sum the signed area of two triangular regions.
Z
4p
(4) Draw a graph of area represented by the definite integral
16 x2 dx. (Hint: think
0
p
about the graph of y = 16 x2 . It’s a familiar geometric shape.) Using geometry, determine the value of the definite integral.
(5) (Bonus question): Prove that
approximations.
Z
1
x2 dx =
0
1
1
by computing the limit of the left-endpoint
3
May
2023
alida
04 May 2023
amindji
"
X
1
=
-
The
is
region
shaded
greater
=
2
=
(bh (t)(3)(4) 6
=
-
-
below
region
positive.
S,f(x)dx (,f(x)dx gf(x)dx ff(x)dx f(x)dx
=
+
+
-
+
S'f(x)dx=Sf(x)dx-Jf(x)dxe -ff(x)dx+)f(x)dX
S2f(x)dx 2)f(x)dX 2(2)
0
=
=
=
=
-
=
32
+
=
-2 3 =
+
4
=
=
-
s-2 6 4
+
As
~
2
sumof
=
mesigned
two
are of
tianquarregions.
--
x
II
"6x2dx=
Area=f
-1
14
(F(2)
=
X-axis
shaded
than the
value is
=
(t)bh (t)(z)(-2)
the
the x-axis. Therefore, the
integral
X 2
gf(x)dx
above
=
4
=
...
=
-
2
4
:
1R 4
=
[f(x)Dx
(ix)ax-Lim?f(xi)sx]
i8
=
2
f(x) x ja 0;b 1jXi a i1X
=
=
1x
=
=
b 10
=
=
=
+
-
x 0;x, 0
=
=
=
=
+
ti
is
ma allnicee
⑭
1(t);x 0 2(t) .....
+
Xi =
Sixdx-m(!t(t)(t)-m!(t)()- m2.()]=m (ne
-((t)) (2-0) = I
Ein]
-
=
=
X2
=
w