In this example we will use Riemann sums to approximate . Recall that the area of the unit circle is, so the area under the graph of f(x)=√1-x² for 0 ≤ x ≤ 1 (and above the x-axis) is precisely /4. 10 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 Figure 2: Quarter Circle 1.0 (a) Compute the left-Riemann sum L5 of f(x) with respect to the partition of [0, 1] into 5 subintervals of equal width. Use this to find an approximation of and explain whether this is an over- or underestimate. (b) Compute the right-Riemann sum Rs of f(x) with respect to the partition of [0, 1] into 5 subintervals of equal width. Use this to find an approximation of and explain whether this is an over- or underestimate.
In this example we will use Riemann sums to approximate . Recall that the area of the unit circle is, so the area under the graph of f(x)=√1-x² for 0 ≤ x ≤ 1 (and above the x-axis) is precisely /4. 10 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 Figure 2: Quarter Circle 1.0 (a) Compute the left-Riemann sum L5 of f(x) with respect to the partition of [0, 1] into 5 subintervals of equal width. Use this to find an approximation of and explain whether this is an over- or underestimate. (b) Compute the right-Riemann sum Rs of f(x) with respect to the partition of [0, 1] into 5 subintervals of equal width. Use this to find an approximation of and explain whether this is an over- or underestimate.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![2. In this example we will use Riemann sums to approximate . Recall that the area of
the unit circle is, so the area under the graph of f(x)=√1-x² for 0≤x≤1 (and
above the x-axis) is precisely π/4.
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
Figure 2: Quarter Circle
1.0
(a) Compute the left-Riemann sum Ls of f(x) with respect to the partition of [0, 1]
into 5 subintervals of equal width. Use this to find an approximation of and
explain whether this is an over- or underestimate.
(b) Compute the right-Riemann sum Rs of f(x) with respect to the partition of [0, 1]
into 5 subintervals of equal width. Use this to find an approximation of and
explain whether this is an over- or underestimate.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F242e2463-ddfc-4cc7-bb20-37587f636c51%2F13a9db9d-257f-4f96-b19f-dcb2a5357539%2Fcauzr8g_processed.png&w=3840&q=75)
Transcribed Image Text:2. In this example we will use Riemann sums to approximate . Recall that the area of
the unit circle is, so the area under the graph of f(x)=√1-x² for 0≤x≤1 (and
above the x-axis) is precisely π/4.
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
Figure 2: Quarter Circle
1.0
(a) Compute the left-Riemann sum Ls of f(x) with respect to the partition of [0, 1]
into 5 subintervals of equal width. Use this to find an approximation of and
explain whether this is an over- or underestimate.
(b) Compute the right-Riemann sum Rs of f(x) with respect to the partition of [0, 1]
into 5 subintervals of equal width. Use this to find an approximation of and
explain whether this is an over- or underestimate.
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