a) Write the sigma notation formula for the right Riemann sum Rn of the function f(x)=4−x2 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=n(n+1)2, ∑k=1nk2=n(n+1)(2n+1)6 b) Use the Fundamental Theorem of Calculus to calculate the derivative of F(x)=∫e−xxln⁡(t2+1)dt

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a) Write the sigma notation formula for the right Riemann sum Rn of the function f(x)=4−x2 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=n(n+1)2, ∑k=1nk2=n(n+1)(2n+1)6 b) Use the Fundamental Theorem of Calculus to calculate the derivative of F(x)=∫e−xxln⁡(t2+1)dt

SabadisO · Accepted Answer

Step 1a) Given: f(x)=4−x2, a=0, b=2, Δx=[b−a]n=(2−0)n=(2n)xi=a+iΔx=0+i(2n)=(2in)f(xi)=4−(2in)2Rn=∑i=1nf(xi)ΔxRn=∑i=1n(4−(2in)2)(2n)∫02f(x)dx=limn≥∞Rn∫02f(x)dx=limn≥∞∑i=1n(4−(2in)2)(2n)∫02f(x)dx=limn≥∞∑i=1n(1−i2n2)(8n)∫02f(x)dx=limn≥∞∑i=1n8((1n)−(i2n3))∫02f(x)dx=limn≥∞8((nn)−n(n+1)(2n+1)6n3)∫02f(x)dx=limn≥∞8(1−(1(1+(1n))2+(1n)6∫02f(x)dx=8×1−1(1+0)(2+0)6∫02f(x)dx=8×(1−26)∫02f(x)dx=8×(23)∫02f(x)dx=(163)Step 2b) ddx∫p(x)q(x)f(t)dt=f(q(x))×q′(x)−f(p(x))×p′(x)F(x)=∫e−xxln⁡(t2+1)dtF′(x)=(ln⁡(x2+1))×(1)−(ln⁡((e−x)2+1))×(−e−x)

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

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