Question

Use C++ to design a program:
Your program will use three different 5-point numerical quadrature methods
(Closed Newton Cotes, Gaussian Quadrature, and Lobatto Quadrature)

You will employ each of these methods to solve each of the following three problems to find the area under the respective curves over the interval (-1, 1).

The following is a plot of the first function f(x) = 1 - sin(1 - x)   

The true area under the curve is (to 20 digits) 0.58385316345285761300.
             

The following is a plot of the second function f(x) = sqrt(x + 1) + 1

The true area under the curve is (to 20 digits) 3.8856180831641267317

The following is a plot of the third function f(x) = tanh(x+1)
The true area under the above curve is (to 20 digits) 1.3250027473578644309

First, print out the values of the weights and the nodes for each quadrature formula,
Then for each function (or curve), print out your approximation to the area under the curve,
the true value of the area, and the error in the approximation for each of the three quadrature formulae.

Note: For Lobatto quadrature, the nodes are x[1] = -1, x[2] = -sqrt(3/7), x[3] = 0, x[4] = sqrt(3/7), and x[5] = 1.
The weights are w[1] = 1/10, w[2] = 49/90, w[3] = 32/45, w[4] = 49/90, and w[5] = 1/10.

Answer 1

• The following is a plot of the first function f(x) = 1 - sin(1 - x)
• The true area under the above curve is (to 20 digits) 0.58385316345285761300

• The following is a plot of the second function f(x) = sqrt(x + 1) + 1

The true area under the above curve is (to 20 digits) 3.8856180831641267317

• The following is a plot of the third function f(x) = tanh(x+1)
• The true area under the above curve is (to 20 digits) 1.3250027473578644309