Find the area above the curve Using Infinite Summation

Question

Find the area above the curve of 3x² – 12x + 9 from [1, 3]. What I know is that the solution would involve infinite summation. The place where I am stuck is the definition of the sum. I know that

Δx= 3-1/n



Answer 1

The Riemann sum approximates the area under a curve by n rectangles, each with height given by evaluating f at increments

f(x₀+kΔx)

and each with width given by

Δx = 2/n

The sum of these is

∑_k f(x₀ + kΔx)(Δx)

where k ranges from k=1 to k= n in the right-side approximation, x₀=1 is the left endpoint, and (as you already found) Δx = (3-1)/n = 2/n. To help remember this setup, it can help to simply draw a picture with the graph of your given function, and rectangles approximating under/above it.

Drawing can be especially helpful when you're asked to set up left or right endpoint approximations and you want to keep track of all of the specific endpoints and the different values for the index k.

Now, evaluate the expression in the summand. Keep in mind that we want to think of k as the changing variable until we can use summation formulas to write the sum in terms of only n. Only after we get rid of the summation and any use of k should we attempt to take the limit.

Forgetting the summation notation for the time being, let's focus on simplifying the summand expression

f(x₀ + k(Δx))(Δx) = (3(1 + 2k/n)² - 12(1 + 2k/n) + 9)(2/n)

= (3(1 + 4k/n + 4k²/n²) - 12 - 24k/n + 9)(2/n)

= -12k/n² + 12k²/n³

Now to sum this over k, treat n as a constant and use the formulas

Σ1 = n (the sum of n 1's, 1 + 1 + ... + 1 = n)

∑k = n(n+1)/2

Σk² = n(n+1)(2n+1)/6

(where the sums are taken from k=1 to k=n)

Answer 2

First, divide the expression by 3 so we have, y= x^2 -4x +3.

Second, find the roots of the quadratic as shown below,

x^2 -4x +3 = 0 or (x-1)*(x-3)=0 then we have x=1 and x=3.

Then, take integration form the quadratic above from 1 to 3 shown below which gives us the area trapped between the equation (above the equation) and x-axis.

ʃ (x^2 -4x +3)= [1/3 (x^3) -2* x^2 +3x](from 1 to 3)= -1.33 (Answer) ( being negative means the area is below the x-axis)