Mat 540 Final Report

Summary:

When integrating function expressed with the new term variable u, we provide the neutralizing factor or “nf” written before the integral sign. This nf could be balance with the given term of the integration.

The antiderivative of the trigonometric, exponential, and logarithmic functions would be based on our knowledge on the differentiation formulas since integration is for the reverse of finding the function.

A separable differential equation is any differential equation that we can write in the following form, N(y)dy/dx = M(x)

Note that in order for a differential equation to be separable all the y's in the differential equation must be multiplied by the derivative and all the x's in the differential equation must be on

∫▒〖x^2 (〖2x〗^3-1)dx〗

2. ∫▒(x+1)dx/∛(x^2+2x+1)

3.∫▒(2x+3)dx/(x^2+3x+4)

4. ∫▒((〖(x〗^(1/3)+1)^(3/2) dx)/x^(2/3)

5.∫▒〖sec x dx〗

6.∫▒〖e^4x dx〗

7. y dx – x2 dy = 0

8. (1 + x2) = dy/dx y3 9. dy/dx=〖sin〗^2 y sin x 10. y dx = x^4 dy

Area of a region and Riemann sum

Review on summation notation which is to be used in the next topic; Activity 6.3.a But first, answer this Evaluate the following notation:

∑_(i=1)^n▒〖(6i+3)〗, when n=4 ∑_(i=1)^n▒〖12i^3 〗, when n = 5 ∑_(i=1)^n▒〖(8i-3)〗, when n = 4 ∑_(i=1)^n▒〖(12i^2+4i)〗, when n = 3 ∑_(i=1)^n▒(3i+1)^3 , when n = 5

Riemann

This technique is called the Riemann Sum. It is named after German Mathematician Bernhard Riemann.

Let us take first these summation notations: The following summation formulas are very useful in evaluating area of a region:

E (1) ∑_(i=1)^n▒〖 i= (n(n+1))/2〗

E (2) ∑_(i=1)^n▒〖i²= (n(n+1)(2n+1))/6〗

E (3) ∑_(i=1)^n▒〖i³= (n² (n+1)²)/4〗

E (4) ∑_(i=1)^n▒〖k=nk〗 k = constant

E (5) ∑_(i=1)^n▒〖ka〗_i = k∑_(i=1)^n▒a_i

E (6) ∑_(i=1)^n▒〖(a_i+ b_i)〗= ∑_(i=1)^n▒a_i + ∑_(i=1)^n▒b_1

We let y = f(x) be a function which is continuous on a closed interval [a, b], and divide this interval into n subintervals (not necessarily equal in lengths). The numbers x_1,x_2, . . xn – 1 such that x_(0 =) a< x_1 < x_2< . . . < x_(n-1)< x_n=b are called points of division. The set of closed subintervals [x_0,x_1 ],[x_1,x_2], . . . , [x_(n-1),x_n], is called a partition of the closed interval [a, b]. Writing the sum in summation notation, we have ∑_(i=1)^n▒〖f(z_i )∆x_i 〗 which is called the Riemann Sum.

These methods of Riemann summation are usually best approached with partitions of equal size. The interval [a, b] is therefore divided into n subintervals.

The three Riemann sum