Left, Right, and Midpoint Riemann Sums
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Problems 11
- What does that Midpoint Riemann sum work out to?
- Order the Left, Right, and Midpoint Riemann sums from smallest to largest.
- What is the left Riemann sum for the function f(x)=6−xf(x) = 6-xf(x)=6−x over the interval [0,8][0, 8][0,8] with 444 rectangles?
- What is the right Riemann sum of the function f(x)=x2−1f(x) = x^2 - 1f(x)=x2−1 over the interval [1,4][1, 4][1,4], using three rectangles?
- What is the midpoint Riemann sum of the function f(x)=sin(x)f(x) = \sin(x)f(x)=sin(x) over the interval [0,π][0, \pi][0,π] using three rectangles?
- Approximate f(x)=∫151x2dxf(x)=\int_1^5\frac{1}{x^2}dxf(x)=∫15x21dx using four rectangles and right endpoints.
- Approximate f(x)=∫151x2dxf(x)=\int_1^5\frac{1}{x^2}dxf(x)=∫15x21dx using four rectangles and left endpoints.
- Approximate f(x)=∫151x2dxf(x)=\int_1^5\frac{1}{x^2}dxf(x)=∫15x21dx using four rectangles and midpoints.
- If you have an increasing function on [a,b][a,b][a,b], which Riemann Sum will underestimate the area under the curve the most?
- If you have a decreasing function on [a,b][a,b][a,b], which Riemann Sum will underestimate the area under the curve the most?
- Evaluate the sum-limit limn→∞∑k=0n−14n(2−4kn)4.\lim_{n \rightarrow \infty}\sum^{n - 1}_{k = 0}\frac{4}{n}\left(2 - \frac{4k}{n}\right)^4.n→∞limk=0∑n−1n4(2−n4k)4.