Calculus help please?

Question

 

If f(x) = 5x^2 − 6x, 0 ≤ x ≤ 3, evaluate the Riemann sum with n = 6, taking the sample points to be right endpoints.  

 

Answer 1

The function f(x) = 5x² - 6x

Number of sub intervals n = 6

Let the f(x) be a function and is continuous on the interval a ≤ x ≤ b.

In this case 0 ≤ x ≤ 3

a = 0, b = 3

Divide this interval into n equal width subintervals, each of which has a width of ∆x = (b - a)/n

∆x = (3 - 0)/6

∆x = 1/2

x₀  = 0

 x₁ = x₀ + ∆x = 0 + 1/2 = 1/2

x₁ = 0.5

x₂ = x₁ + ∆x = 1/2 + 1/2 = 1

x₂ = 1

x₃ = x₂ + ∆x = 1/4 + 1/2 = 3/4

x₃ = 0.75

x₄ = x₃ + ∆x = 3/4 + 1/2 = 5/4

x₄  = 1.25

x₅ = x₄ + ∆x = 5/4 + 1/2 = 7/4

x₅  = 1.75

x₆ = x₅ + ∆x = 7/4 + 1/2 = 9/4

x₆  = 2.25

f(x) = 5x² - 6x

f(0.5) = 5(0.5)² - 6(0.5) =  - 1.75

f(1) = 5(1)² - 6(1) = 5 - 6 = - 1

f(0.75) = 5(0.75)² - 6(0.75) =  - 1.6875

f(1.25) = 5(1.25)² - 6(1.25) =  0.3125

f(1.75) = 5(1.75)² - 6(1.75) = 4.8125

f(2.25) = 5(2.25)² - 6(2.25) =  11.8125

The Riemann sum is 

So the Riemann sum is

R₆ = f(0.5)∆x + f(1)∆x + f(0.75)∆x + f(1.25)∆x + f(1.75)∆x + f(2.25)∆x

= ∆x[ f(0.5) + f(1) + f(0.75) + f(1.25) + f(1.75) + f(2.25)]

= 1/2(- 1.75 - 1 - 1.6875 + 0.3125 + 4.8125 + 11.8125)

= 12.5/2

R₆ = 6.25.

Comments

this answer is WRONG!

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Yes, you are right and the answer is wrong.

Please check my answer below.

Answer 2

Step 1:

The function is and the interval is .

The number of subintervals are .

The width is .

Substitute and .

.

The Riemann sum is .

Find values.

Step 2:

Find the function values .

Substitute , and values in .

The Riemann sum is .

Solution:

The Riemann sum is .