Math 106
Quiz 2
9/21/12
Solutions
π/2
Z
1. Let I =
x sin x dx. How many subdivisions are required to obtain a left sum approximation
0
with error of at most 1/10,000?
1.4
Since f (x) = x sin x, then f 0 (x) = sin x+x cos x.
To find K1 we need to find the maximum value
of | sin x + x cos x| on [0, π2 ]. Looking at the
graph reveals that |f 0 (x)| achieves a maximum
value of approximately 1.391008 on [0, π/2], so
let K1 = 1.4. The error bound estimates for left
sums may be determined using:
K1 (b − a)2
.
|I − Ln | ≤
2n
1.2
| f’ |
1.0
0.8
0.6
0.4
0.2
1.0
0.5
Therefore,
K1 (b − a)2
1
≤
2n
10000
⇐⇒
2
1.4 π2 − 0
1
≤
2n
10000
⇐⇒
n≥
14000π 2
8
1.5
≈ 17271.81.
Therefore, we require (at least) n = 17272 subdivisions.
2. To find the area of the region between y = 2x2
and y = x4 − 2x2 , first find the x-coordinates for
the intersection points:
y = x4 − 2x2
2x2 = x4 − 2x2 ⇐⇒ x4 − 4x2 = 0
2
y = 2x2
2
⇐⇒ x (x − 4) = 0
⇐⇒ x2 (x + 2)(x − 2) = 0
⇐⇒ x = −2, 0, 2
-2
0
Notice that 2x2 ≥ x4 − 2x2 on [−2, 2], therefore we may view y = 2x2 as the “top” function and
y = x4 − 2x2 as the “bottom” function. In addition, the shaded region is symmetric across the
y−axis, therefore we can find the area on the right hand side (i.e., from x = 0 to x = 2) and double
it to find the desired area. The area of the shaded region may be found using the integral below:
Z
2
0
2
2
4
1
[2x − (x − 2x )] dx = 2
(4x − x ) dx = 2 x3 − x5
3
5
0
4 3 1 5
=2
(2) − (2) − 0
3
5
32 32
128
=2
−
=
3
5
15
2
4
2
Z
2
1
4
2
0
2