Estimate the area. Riemann Sum

Question

(a) Estimate the area under the graph of the function f(x)=1x+8 from x=0 to x=1 using a Riemann sum with  n=10 subintervals and right endpoints.

 

Round your answer to four decimal places.

 

area = ____________

 

(b) Estimate the area under the graph of the function f(x)=1x+8 from x=0 to x=1 using a Riemann sum with  n=10 subintervals and left endpoints.

 

Round your answer to four decimal places.

 

area = ____________

Answer 1

First remember that a Riemann sum estimates the area under a curve by dividing the interval in rectangles on equal width.

The internal is from x ∈ [0,1] and we need to dived it ten times. Therefore the width of our 10 is rectangles

w = (1-0)/10 = 0.1, w = width

So the edges of the rectangles are positioned at x_i = [0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]

we have that the height of our rectangle will be given by

f(x) = x + 8

The question is the height is given by the right edges of the rectangles or the left edges

Right edges: x ∈ {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}

Left edges: x ∈ {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}

So now we can calculate the heights by evaluating the right edges on f(x), we do the same for the Left edges

heights using the right edges: h ∈ {8, 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9}

heights using the left edges: h ∈ {8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, 8.9, 9.0}

So now we can multiply the heights by their width to get the area of each rectangle and sum them all to get the total area.

A = (0.8 + 0.81 + 0.82 + 0.83 + 0.84 + 0.85 + 0.86 + 0.87 + 0.88 + 0.89), using right edges

A = (0.81 + 0.82 + 0.83 + 0.84 + 0.85 + 0.86 + 0.87 + 0.88 + 0.89 + 0.90), using left edges

Total A = 8.45, using right edges

Total A = 8.55, using left edges

Answer 2

(a) Δx = (1 - 0)/10 = 0.1. Then right Riemann sum R₁₀ = ∑₁¹⁰(01.i + 8)·0.1 = 8.55

(b) Δx = (1 - 0)/10 = 0.1. Left Riemann sun L₁₀ = ∑₀⁹(01.i + 8)·0.1 = 8.45

Fo r calculation I used TI-84 sum(seq) command of List menu.

Answer 3

a. Right end points of RIemann sum means for each rectangle sub-interval the right leg hits the function.

In this problem, n=10 which means Delta x = (1 - 0)/10 = 0.1; so the x coordinates of the right end points are:

0.1 , 0.2 , 0.3 , 0.4 , 0.5 , 0.6 , 0.7 , 0.8 , 0.9 and 1.0

Plug in these x-coordinates into the function f(x)=1x+8, so

f(0.1)=8.1 , f(0.2)=8.2 , etc

The right end points Riemann sum = 0.1 * (8.1 + 8.2 + 8.3 + ... + 9)= 8.55

b. Left end points: same Delta x = 0.1 ; the x coordinates of the left end points are:

0, 0.1, 0.2, ... 0.9

f(0) = 8. , f(0.1)=8.1 , f(0.2)=8.2 ... f(0.9)=8.9

The left end points RIemann sum = 0.1 * (8+8.1+8.2+...+8.9) = 8.45

Of course we can simply use geometry to calculate ACTUAL area of the trapezoid shape region: ((8+9) * 1)/2= 8.5

or using Integral, the anti derivative is (x^2)/2 + 8x and plug in 1 as upper limit and 0 as lower limit yields 8.5 as well.