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=n(n+1)/2,\ \sum_{k=1}^{n}k^{2}=n(n+1)(2n+1)/6)

djeljenike 2021-08-20 Answered
a) Write the sigma notation formula for the right Riemann sum Rn of the function f(x)=4x2 on the interval [0, 2] using n subintervals of equal length, and calculate the definite integral 02f(x)dx as the limit of Rn at n.
(Reminder: k=1nk=nn+12, k=1nk2=n(n+1)2n+16)
b) Use the Fundamental Theorem of Calculus to calculate the derivative of F(x)=exxln(t2+1)dt
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Expert Answer

lamusesamuset
Answered 2021-08-21 Author has 93 answers
Step 1
a) Given: f(x)=4x2, a=0, b=2, Δx=ban=(20)n=(2n)
xi=a+iΔx=0+i(2n)=(2in)
f(xi)=4(2in)2
Rn=i=1nf(xi)Δx
Rn=i=1n(4(2in)2)(2n)
02f(x)dx=limnRn
02f(x)dx=limni=1n(4(2in)2)(2n)
02f(x)dx=limni=1n(1(i2n2))(8n)
02f(x)dx=limni=1n8((1n)(i2n3))
02f(x)dx=limn8((nn)(n(n+1)2n+16n3))

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