Question

# a) Write the sigma notation formula for the right Riemann sum R_{n} of the function

Integrals

a) Write the sigma notation formula for the right Riemann sum $$R_{n}$$ of the function $$f(x)=4-x^{2}$$ on the interval $$[0,\ 2]$$ using n subintervals of equal length, and calculate the definite integral $$\int_{0}^{2}f(x) dx$$ as the limit of $$R_{n}$$ at $$n\rightarrow\infty$$.
(Reminder: $$\sum_{k=1}^{n}k=\frac{n(n+1)}{2},\ \sum_{k=1}^{n}k^{2}=\frac{n(n+1)(2n+1)}{6}$$
b) Use the Fundamental Theorem of Calculus to calculate the derivative of $$F(x)=\int_{e^{-x}}^{x}\ln(t^{2}+1)dt$$

2021-05-15

Step 1
a) Given: $$f(x)=4-x^{2},\ a=0,\ b=2,\ \Delta x=\frac{[b-a]}{n}=(2-0)_{n}=(\frac{2}{n})$$
$$x_{i}=a+i\Delta x=0+i(\frac{2}{n})=(\frac{2i}{n})$$
$$f(x_{i})=4-(\frac{2i}{n})^{2}$$
$$R_{n}=\sum_{i=1}^{n}f(x_{i})\Delta x$$
$$R_{n}=\sum_{i=1}^{n}(4-(\frac{2i}{n})^{2})(\frac{2}{n})$$
$$\int_{0}^{2}f(x)dx=\lim_{n\geq\infty}R_{n}$$
$$\int_{0}^{2}f(x) dx=\lim_{n\geq\infty}\sum_{i=1}^{n}(4-(\frac{2i}{n})^{2})(\frac{2}{n})$$
$$\int_{0}^{2}f(x) dx=\lim_{n\geq\infty}\sum_{i=1}^{n}(\frac{1-i^{2}}{n^{2}})(\frac{8}{n})$$
$$\int_{0}^{2}f(x) dx=\lim_{n\geq\infty}\sum_{i=1}^{n}8((\frac{1}{n})-(\frac{i^{2}}{n^{3}}))$$
$$\int_{0}^{2}f(x) dx=\lim_{n\geq\infty}8((\frac{n}{n})-\frac{n(n+1)(2n+1)}{6n^{3}})$$
$$\int_{0}^{2}f(x) dx=\lim_{n\geq\infty}8(1-(1(1+(\frac{1}{n}))\frac{2+(\frac{1}{n})}{6}$$
$$\int_{0}^{2}f(x) dx=8\times1-\frac{1(1+0)(2+0)}{6}$$
$$\int_{0}^{2}f(x) dx=8\times(1-\frac{2}{6})$$
$$\int_{0}^{2}f(x) dx=8\times(\frac{2}{3})$$
$$\int_{0}^{2}f(x) dx=(\frac{16}{3})$$
Step 2
b) $$\frac{d}{dx}\int_{p(x)}^{q(x)}f(t)dt=f(q(x))\times q'(x)-f(p(x))\times p'(x)$$
$$F(x)=\int_{e^{-x}}^{x}\ln(t^{2}+1)dt$$
$$F'(x)=(\ln(x^{2}+1))\times(1)-(\ln((e^{-x})^{2}+1))\times(-e^{-x})$$
$$F'(x)=\ln(x^{2}+1)+e^{-x}\ln(e^{-2x}+1)$$