Calculate the upper sums Un and lower sums Ln , on a regular partition of the intervals, for the following integrals.

Question

Calculate the upper sums Un and lower sums Ln, on a regular partition of the intervals, for the following integrals.

(a)  ∫7 1 (2−8x)dx

(b)  ∫1 0 (4+18x2)dx

(c)  ∫2 1 H(x−2)dx where H(x) is the Heaviside function

(d)  ∫2 1 D(x)dx where D(x) is the Dirichlet function

Answer 1

Answer 2

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Let's calculate the upper sums (Un) and lower sums (Ln) on regular partitions for each of the given integrals:

**(a) ∫7 to 1 (2 - 8x)dx**

First, we need to find the partition of the interval [1, 7] into subintervals with equal width (Δx).

Let's choose n subintervals, so Δx = (7 - 1) / n = 6/n.

Now, we'll calculate the upper and lower sums for this integral using the regular partition.

Upper Sum (Un):

Un = Σ[Δx * f(xi)] for i = 1 to n

Un = Σ[(6/n) * (2 - 8xi)] for i = 1 to n

We can simplify this by factoring out the (6/n):

Un = (6/n) * Σ[2 - 8xi] for i = 1 to n

Now, calculate the sum:

Un = (6/n) * [2n - 8(1) - 8(2) - ... - 8(n-1)]

Lower Sum (Ln):

Ln = Σ[Δx * f(xi)] for i = 1 to n

Ln = Σ[(6/n) * (2 - 8xi)] for i = 1 to n

Again, factor out the (6/n):

Ln = (6/n) * Σ[2 - 8xi] for i = 1 to n

Now, calculate the sum:

Ln = (6/n) * [2(1) + 2(2) + ... + 2(n-1) - 8(1) - 8(2) - ... - 8(n-1)]

**(b) ∫1 to 0 (4 + 18x^2)dx**

Follow the same procedure to calculate Un and Ln for this integral.

**(c) ∫2 to 1 H(x - 2)dx where H(x) is the Heaviside function**

The Heaviside function H(x) is defined as:

H(x) = {

0, if x < 0,

1, if x ≥ 0

}

In this case, the integral ∫2 to 1 H(x - 2)dx will evaluate to:

∫2 to 1 H(x - 2)dx = ∫2 to 1 0 dx = 0

Both Un and Ln will be 0 for this integral because H(x - 2) is 0 in the given interval.

**(d) ∫2 to 1 D(x)dx where D(x) is the Dirichlet function**

The Dirichlet function D(x) is defined as:

D(x) = {

1, if x is rational,

0, if x is irrational

}

In this case, the integral ∫2 to 1 D(x)dx will evaluate to:

∫2 to 1 D(x)dx = ∫2 to 1 0 dx = 0

Both Un and Ln will be 0 for this integral because D(x) is 0 in the given interval.

So, for the Dirichlet and Heaviside functions, the upper and lower sums are both 0 because the functions are zero over the interval of integration.