Question

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 3 sin(x2) dx, n

Answer 1

The value of the given statement according to the mid point theorem is 5.

According to the statement

we have given that the equation and we have to integrate it with the help of the mid point theorem.

So, For this purpose, we know that the

The midpoint theorem states that “The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”

So, The given equation is

$\int\limits^6_0 {Sinx } \, dx$

where n = 4

And

The mid point formula is a with limits a to b is

f(x) dx ≈ Δx (f (x₀ + x₁)/2) + (f (x₁ + x₂)/2) + ... (f (xₓ₋₂ + xₓ₋₁)/2) + (f (xₓ₋₁ + x)/2))

Then

Where Δx = (b-a)/n

Recall that

a= 0

b= 64 and

n=4, therefore,

Δx = (64-0)/4 = 16

The next step requires that the interval [0,64] be divided into 4 sub-intervals with length = Δx =16

Hence, we've got, 0, 16, 32, 48, 64.

From this point, we calibrate the respective functions as follows:

f (x₀ + x₁)/2) = f ((0+16)/2) = f(8) = Sin (8) = 0.98935824662

(f (x₁ + x₂)/2) = f((16+32)/2) = f(24) =Sin (24) = -0.905578362

(f (x₂ + x₃)/2) = f((32+48)/2) = f(40) = Sin (40) = 0.74511316047

(f (x₃ + x₄)/2) = f((48+64)/2) = f(56) = Sin (56) = -0.52155100208

At this point, we sum up the above values derive the product of the total and Δx = 16

sin (x) dx = 16(0.98935824662 - 0.905578362 +  0.74511316047 - 0.52155100208)

= 16 (0.30734204301)

= 4.91747268816

$\int\limits^6_0 {Sinx } \, dx = 5$

So, The value of the given statement according to the mid point theorem is 5.

Learn more about Mid point theorem here

brainly.com/question/9635025

#SPJ4