Find the value of V

Question

Find the value V of the Riemann sum V= ∑n,k=1 ƒ(ck)Δx_{k, for the functionƒ(x)=x}² +4 using the partition P=(1,3,4,6), where c_k is the left endpoints of the partition.

Answer 1

f(x) = x² + 4

Using left endpoints, the first approximating rectangle has height f(1) = 5 and width Δx₁ = 3-1 = 2

The second approximating rectangle has height f(3) = 13 and width Δx₂ = 4-3 = 1.

The third approximating rectangle has height f(4) = 20 and width Δx₃ = 6-4 = 2

So, the approximating sum is f(1)Δx₁ + f(3)Δx₂ + f(4)Δx₃ = 10 + 13 + 40 = 63

Answer 2

f(1) = 5

f(3) = 13

f(4) = 20

f(6) = 40

2*5 + 1 * 13 + 2*20 + 40 = 10 + 13 + 40 + 40 = 103

Next, integrating: (1/3)x^3 + 4x

(1/3)*{6*6*6} + 4(6) - (1/3)(1*1*1) - 4*1 =

(1/3)*216 + 24 - 1/3 - 4 =

72 + 24 - (4 and 1/3) =

98 - 4 and 1/3 =

93 and 2/3