Estimate area, 4 left endpoints and rectangles. Is this an overestimate or underestimate?
\(\displaystyle f(x) = 4 \cos x\) on interval \(\displaystyle [0,\dfrac{\pi}{2}]\) given \(\displaystyle [a, b]\)
\(\displaystyle \sum\limits_{i=4}^n \Delta x [f(a + i \Delta x - \Delta x)]\)
\(\displaystyle \Delta x = \dfrac{b - a}{n}\)
\(\displaystyle \Delta x = \dfrac{\dfrac{\pi}{2} - 0}{4} = \dfrac{\pi}{8}\)
\(\displaystyle n = 4\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0 +(1)(\dfrac{\pi}{8}) - (\dfrac{\pi}{8}))] + [f(0 +(2)(\dfrac{\pi}{8}) - (\dfrac{\pi}{8})] + [f(0 +(3)(\dfrac{\pi}{8}) - (\dfrac{\pi}{8}))] + [f(0 +(4)(\dfrac{\pi}{8}) - (\dfrac{\pi}{8}))]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0 + (\dfrac{\pi}{8}) - (\dfrac{\pi}{8}))] + [f(0 + (\dfrac{2\pi}{8}) - (\dfrac{\pi}{8}))] + [f(0 + (\dfrac{3\pi}{8}) - (\dfrac{\pi}{8}))] + [f(0 + (\dfrac{4\pi}{8}) - (\dfrac{\pi}{8}))]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0 + (\dfrac{\pi}{8}) - (\dfrac{\pi}{8}))] + [f(0 + (\dfrac{\pi}{4}) - (\dfrac{\pi}{8})] + [f(0 + (\dfrac{3\pi}{8}) - (\dfrac{\pi}{8}))] + [f(0 + (\dfrac{\pi}{2}) - (\dfrac{\pi}{8}))]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0 + 0)] + [f(0 + \dfrac{\pi}{8})] + [f(0 + \dfrac{2\pi}{8})] + [f(0 + \dfrac{3\pi}{8})]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0 + 0)] + [f(0 + \dfrac{\pi}{8})] + [f(0 + \dfrac{\pi}{4})] + [f(0 + \dfrac{3\pi}{8})]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0)] + [f(\dfrac{\pi}{8})] + [f(\dfrac{\pi}{4})] + [f( \dfrac{3\pi}{8})]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[4 \cos (0)] + [4 \cos (\dfrac{\pi}{8})] + [4 \cos (\dfrac{\pi}{4})] + [4 \cos ( \dfrac{3\pi}{8})]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[4 (1)] + [4(.9239)] + [4 (.7071)] + [4(.3827)]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[4] + [3.6972] + [2.8284] + [1.5308]]\)
\(\displaystyle \sum\limits_{i=4}^n (.3927) [[4] + [3.6972] + [2.8284] + [1.5308]]\)
\(\displaystyle \sum\limits_{i=4}^n [1.5708] + [1.4519] + [1.1107] + [.6011]]\) - on the right track?
\(\displaystyle f(x) = 4 \cos x\) on interval \(\displaystyle [0,\dfrac{\pi}{2}]\) given \(\displaystyle [a, b]\)
\(\displaystyle \sum\limits_{i=4}^n \Delta x [f(a + i \Delta x - \Delta x)]\)
\(\displaystyle \Delta x = \dfrac{b - a}{n}\)
\(\displaystyle \Delta x = \dfrac{\dfrac{\pi}{2} - 0}{4} = \dfrac{\pi}{8}\)
\(\displaystyle n = 4\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0 +(1)(\dfrac{\pi}{8}) - (\dfrac{\pi}{8}))] + [f(0 +(2)(\dfrac{\pi}{8}) - (\dfrac{\pi}{8})] + [f(0 +(3)(\dfrac{\pi}{8}) - (\dfrac{\pi}{8}))] + [f(0 +(4)(\dfrac{\pi}{8}) - (\dfrac{\pi}{8}))]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0 + (\dfrac{\pi}{8}) - (\dfrac{\pi}{8}))] + [f(0 + (\dfrac{2\pi}{8}) - (\dfrac{\pi}{8}))] + [f(0 + (\dfrac{3\pi}{8}) - (\dfrac{\pi}{8}))] + [f(0 + (\dfrac{4\pi}{8}) - (\dfrac{\pi}{8}))]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0 + (\dfrac{\pi}{8}) - (\dfrac{\pi}{8}))] + [f(0 + (\dfrac{\pi}{4}) - (\dfrac{\pi}{8})] + [f(0 + (\dfrac{3\pi}{8}) - (\dfrac{\pi}{8}))] + [f(0 + (\dfrac{\pi}{2}) - (\dfrac{\pi}{8}))]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0 + 0)] + [f(0 + \dfrac{\pi}{8})] + [f(0 + \dfrac{2\pi}{8})] + [f(0 + \dfrac{3\pi}{8})]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0 + 0)] + [f(0 + \dfrac{\pi}{8})] + [f(0 + \dfrac{\pi}{4})] + [f(0 + \dfrac{3\pi}{8})]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[f(0)] + [f(\dfrac{\pi}{8})] + [f(\dfrac{\pi}{4})] + [f( \dfrac{3\pi}{8})]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[4 \cos (0)] + [4 \cos (\dfrac{\pi}{8})] + [4 \cos (\dfrac{\pi}{4})] + [4 \cos ( \dfrac{3\pi}{8})]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[4 (1)] + [4(.9239)] + [4 (.7071)] + [4(.3827)]]\)
\(\displaystyle \sum\limits_{i=4}^n (\dfrac{\pi}{8}) [[4] + [3.6972] + [2.8284] + [1.5308]]\)
\(\displaystyle \sum\limits_{i=4}^n (.3927) [[4] + [3.6972] + [2.8284] + [1.5308]]\)
\(\displaystyle \sum\limits_{i=4}^n [1.5708] + [1.4519] + [1.1107] + [.6011]]\) - on the right track?
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