Given f(x) cos(2x) and the interval [0, 3 frac{pi}{4}] approximate the area bounded by the graph of f(x) and the axis on the interval using a left, right, and mid point Riemann sum with n = 3

remolatg 2021-01-28 Answered
Given f(x) cos(2x) and the interval
[0, 3 π4]
approximate the area bounded by the graph of f(x) and the axis on the interval using a left, right, and mid point Riemann sum with n=3
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Expert Answer

dessinemoie
Answered 2021-01-29 Author has 90 answers

Step 1
Interval [0, 3 π4] is to be devided into 3 sub-intervals whos length is given by
h= b  an=3π4  03=π4
Sep 2
Therefore, the three sub-intervals are
[0, π4], [π4, 2π4][2π4, 3π4]
Step 3
Area bounded by the graph of f(x) and the axis on the interval using a left end point Riemann sum id given by
AL=h[|f(0)| + |f(π4)| + |f(2π4)|]
=π4[|cos 0| + |cos π2| + |cos π|]=π4[1 + 0 + 1]=π2
Step 4
Area bounded by the graph of f(x) and the axis on the interval using a right end point Riemann sum is given by
Ar=h[|f(π4)| + |f(2π4)| + |f(3π4)|]
=π4[|cos × π2| + |cos π| + |cos × 3π2|]=π4[0 + 1 + 0]=π4
Step 5
Area bounded by the graph of f(x) and the axis on the interval using a mid point Riemann sum is given by
Am=[|f(π8)| + |f(3π8)| + |f(5π8)|]
=π4[|cos × π4| + |cos × 3π4| + |cos × 5π4|]=π4[12 + 12 + 12]=3π42

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