Question

Derive the formula for Trapezoidal Rule by using Geometry.​

Answer 1

Answer:

this is to explain in short form:

Step-by-step explanation:

We write the Trapezoidal Rule formula with n=3 subintervals: T3=Δx2[f(x0)+2f(x1)+2f(x2)+f(x3)]. f(x2)=f(43)=(43)2=169; f(x3)=f(2)=22=4.

Answer 2

Answer:

Trapezoidal Rule Formula

Then the Trapezoidal Rule formula for area approximating the definite integral \int_{a}^{b}f(x)dx is given by:

When n →∞, R.H.S of the expression approaches the definite integral \int_{a}^{b}f(x)dx.

\int_{1}^{5}\sqrt{1+x^{2}}dx.

Hence, \int_{1}^{5}\sqrt{1+x^{2}}dx ≈ 12.76.

Step-by-step explanation:

Example 1:

Use the trapezoidal rule with n = 8 to estimate:

∫511+x2−−−−−√dx

Solution:

Given, function: ∫511+x2−−−−−√dx

we know that, a=1, b=5 and n=8.

Now, substitute the values in the formula, we get

Δx = (b-a)/n

Δx = (5-1)/8

Δx = 1/2

Now, divide the interval into 8 subintervals with the length of Δx = 1/2, with the following endpoints,

a=1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, 5 = b

Now, compute the functions with these endpoints,

f(x0) = f(1) = √2 = 1.4142135623731

2f(x1) = 2f(3/2) = √13 = 3.60555127546399

2f(x2) = 2f(2) = 2√5 = 4.47213595499958

2f(x3) = 2f(5/2) = √29 = 5.3851648071345

2f(x4) = 2f(3) = 2√10 = 6.32455532033676

2f(x5) = 2f(7/2) = √53 = 7.28010988928052

2f(x6) = 2f(4) = 2√17 = 8.24621125123532

2f(x7) = 2f(9/2) = √85 = 9.21954445729289

2f(x8) = 2f(5) = √26 = 5.09901951359278

Now, substitute the values in the trapezoidal rule formula,

∫baf(x)dx≈Tn=△x2[f(x0)+2f(x1)+2f(x2)+….2f(xn−1)+f(xn)]

= 1/4 (1.4142135623731 + 3.60555127546399 + 4.47213595499958 + 5.3851648071345 + 6.32455532033676 +7.28010988928052 + 8.24621125123532 + 9.21954445729289 + 5.09901951359278)

= 1/4( 51.0465060317)

= 12.7616265079

Which can be approximately written as 12.76

Hence, ∫511+x2−−−−−√dx ≈ 12.76