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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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

dessinemoie · Accepted Answer

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

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

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