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# The following questions are based on the area under the curve for f(x) = x^2 - 3

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The following questions are based on the area under the curve for f(x) = x2 - 3 on the interval 1 ≤ x ≤ 5 with four subintervals.

(a) Find an expression for xk.

(b) Find an expression for f(xk).

(c) Express the area under the curve in summation notation with Right Endpoints.

(d) Calculate the area under the curve with Right Endpoints.

(e) Calculate the area under the curve with Left Endpoints.

(f) Is L4 an underestimate or an overestimate?

(g) Is R4 an underestimate or an overestimate?

Jan 14, 2020

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Here's my best effort, GM

(a)    xi  =   1 + n*Δx         where  Δx  =   [ 5 - 1 ] / 4  =  1

So

xi = 1 + n(1)  =  1 + n

(b)  f(xi )  =   ( 1 + n)^2 - 3

( c)   and (d)  = Right Endpoint Estimate

4                               4

∑     f(xi)  Δx   =        ∑       (1 + n)^2 - 3      =  42  units^2

n = 1                        n = 1

(e)   Left Endpoint Estimate

3                               3

∑     f(xi)  Δx   =        ∑       (1 + n)^2 - 3      =  18  units^2

n = 0                        n = 0

Using Calculus, the true area  is  29 + 1/3  units^2

(f) L4  is an underestimate

(g) R4  is an overestimate   Jan 15, 2020