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For the function given below, find a formula for the Riemann sum obtained by dividing the interval [1,4] into n equal subintervals and using the right-hand endpoint for each ck. Then take a limit of this sum as n -> infinite to calculate the area under the curve over [1: 4].
f(x) = 4x

Sagot :

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Answer:

[tex]\sum \limits ^{n}_{k=1} 4 \Big [ 1 + \dfrac{3k}{n} \Big] \Big [ \dfrac{3}{n} \Big ][/tex]

Step-by-step explanation:

Given the function:

f(x) = 4x; we are to determine the expression given the Reimman sum formula for the given function f(x) = 4x over the interval [1,4]

Since;

[tex]\Delta x = \dfrac{4-1}{x} = \dfrac{3}{x} \\ \\ x_i = a+ \Delta x_i[/tex]

where;

a = 1 and Δ = 4

∴

[tex]x_i = 1+ \dfrac{3}{x}i[/tex]

For i = k

[tex]x_k = 1+ \dfrac{3}{x}k[/tex]

However;

[tex]y(x_i) = 4x \\ \\ y(x_i) = 4(1 +\dfrac{3i}{x})[/tex]

Thus, the formula for the Reinmann sum is:

[tex]\sum \limits ^{n}_{k=1} \Big [ 4 \Big [ 1 + \dfrac{3i}{x} \Big] \Big ] \Delta x \\ \\ \\ \sum \limits ^{n}_{k=1} \Big [ 4 \Big [ 1 + \dfrac{3k}{x} \Big] \Big ] \dfrac{3}{x}[/tex]

Since we are taking the limit as n → ∞

[tex]\sum \limits ^{n}_{k=1} 4 \Big [ 1 + \dfrac{3k}{n} \Big] \Big [ \dfrac{3}{n} \Big ][/tex]

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