Question

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

Integrals
ANSWERED
asked 2021-05-14

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 \(F(x)=\int_{e^{-x}}^{x}\ln(t^{2}+1)dt\)

Answers (1)

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) \(\frac{d}{dx}\int_{p(x)}^{q(x)}f(t)dt=f(q(x))\times q'(x)-f(p(x))\times p'(x)\)
\(F(x)=\int_{e^{-x}}^{x}\ln(t^{2}+1)dt\)
\(F'(x)=(\ln(x^{2}+1))\times(1)-(\ln((e^{-x})^{2}+1))\times(-e^{-x})\)
\(F'(x)=\ln(x^{2}+1)+e^{-x}\ln(e^{-2x}+1)\)

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