write-the-left-and-right-riemann-sums-in-sigma-notation-for-an-arbitrary-value-o

Question

Write the left and right Riemann sums in sigma notation for an arbitrary value of nn to approximate the definite integral ∫241+x2 dx\$$int_2^4$$\sqrt{1 + $$x^2$$}\,dx.

Accepted Answer

Left: ∑i=1n1+(2+2(i−1)n)2⋅2n\sum_{i=1}^{n}\sqrt{1+\left(2+\frac{2\left(i-1\right)}{n}\$$right)^2$$}\cdot\frac{2}{n} Right: ∑i=1n1+(2+2in)2⋅2n\sum_{i=1}^{n}\sqrt{1+\left(2+\frac{2i}{n}\$$right)^2$$}\cdot\frac{2}{n}

Answer

Left: ∑i=1n1+(2+2in)2⋅2n\sum_{i=1}^{n}\sqrt{1+\left(2+\frac{2i}{n}\$$right)^2$$}\cdot\frac{2}{n} Right: ∑i=1n1+(2+2(i−1)n)2⋅2n\sum_{i=1}^{n}\sqrt{1+\left(2+\frac{2\left(i-1\right)}{n}\$$right)^2$$}\cdot\frac{2}{n}

Answer

Left: ∑i=1n1+(2+i−1n)2⋅1n \sum_{i=1}^{n} \sqrt{1 + \left(2 + \frac{i-1}{n}\$$right)^2$$} \cdot \frac{1}{n}  Right: ∑i=1n1+(2+in)2⋅1n \sum_{i=1}^{n} \sqrt{1 + \left(2 + \frac{i}{n}\$$right)^2$$} \cdot \frac{1}{n}

Answer

Left: ∑i=1n1+(i−1n)2⋅2n \sum_{i=1}^{n} \sqrt{1 + \left(\frac{i-1}{n}\$$right)^2$$} \cdot \frac{2}{n}  Right: ∑i=1n1+(in)2⋅2n \sum_{i=1}^{n} \sqrt{1 + \left(\frac{i}{n}\$$right)^2$$} \cdot \frac{2}{n}

Write the left and right Riemann sums in sigma notation for an arbitrary value of nn to approximate the definite integral ∫241+x2dx\(\int\)_2^4\(\sqrt{1 + x^2}\)\,dx.

Write the left and right Riemann sums in sigma notation for an arbitrary value of $n$ to approximate the definite integral $\int_{2}^{4} \sqrt{1 + x^{2}} d x$ .