Integration Using Euler's Method
Key Questions

Estimating Definite Integral by Euler's Method
Example
Use Euler's Method to approximate the definite integral
#int_{1}^2(4x^2)dx# .For simplicity, let us use the step size
#Deltax=1# .Let
#I(t)=int_{1}^t(4x^2)dx# .So, we wish to approximate
#I(2)=int_{1}^2(4x^2)dx# Note that by Fundamental Theorem of Calculus I,
#I'(t)=4t^2# Now, let us start approximating.
#I(1)=\int_{1}^{1}(4x^2)dx=0# By linear approximation,
#I(0) approx I(1)+I'(1)cdot Delta x=0+3cdot1=3# #I(1) approx I(0)+I'(0)cdot Delta x approx3+4cdot1=7# #I(2) approx I(1)+I'(1)cdot Delta x approx 7+3cdot1=10# Hence,
#I(2)=int_{1}^2(4x^2)dx approx 10#
I hope that this was helpful.

To approximate an integral like
#\int_{a}^{b}f(x)\ dx# with Euler's method, you first have to realize, by the Fundamental Theorem of Calculus, that this is the same as calculating#F(b)F(a)# , where#F'(x)=f(x)# for all#x\in [a,b]# . Also note that you can take#F(a)=0# and just calculate#F(b)# .In other words, since Euler's method is a way of approximating solutions of initialvalue problems for firstorder differential equations, we want to calculate
#\int_{a}^{b}f(x)\ dx=F(b)# by approximating the unique solution of#dy/dx=f(x), y(a)=0# at#x=b# .This can be done with an "iteration scheme". Pick a positive integer
#n# to be the number of steps of Euler's method you want to use and then let#Delta x=(ba)/n# . Once this is done, let#x_{0}=a# and#y_{0}=0# and use the recursive equations#x_{k+1}=x_{k}+Delta x# ,#y_{k+1}=y_{k}+Delta y=y_{k}+f(x_{k})\cdot Delta x# to generate a sequence of#n+1# points#(x_{0},y_{0}), (x_{1}, y_{1}), \ldots (x_{n},y_{n})# that approximate the unique solution of the initialvalue problem.The final result is that
#\int_{a}^{b}f(x)\ dx=F(b)\approx y_{n}# .This can be implemented fairly easily on a calculator or computer, though you'd have to be somewhat experienced with such programming.
As an example, suppose that you want to estimate
#\int_{0}^{3}x^{2}\ dx# (which we already know is 9). The relevant initialvalue problem is#dy/dx=f(x)=x^2, y(0)=0# and we want to approximate#y(3)# . Let's choose#n=4# so that#Delta x=\frac{3}{4}=0.75# . Then#x_{0}=0, x_{1}=0.75, x_{2}=1.5, x_{3}=2.25# , and#x_{4}=3# . Also#y_{1}=y_{0}+f(0)\cdot 0.75=0+0=0# ,#y_{2)=y_{1}+f(0.75)\cdot 0.75=0+0.5625\cdot 0.75=0.421875# ,#y_{3}=y_{2}+f(1.5)\cdot 0.75=0.421875+2.25\cdot 0.75=2.109375# , and#y_{4}=y_{3}+f(2.25)\cdot 0.75=2.109375+3.796875=5.90625# as our approximate answer for the integral.This would get to be a better approximation (though very slowly) as
#n# increases (and#Delta x=(ba)/n# decreases). For instance, if#n=100# and#Delta x=3/100=0.03# , the approximation for the integral is#8.86545# . 
Estimating Definite Integral by Euler's Method
Example
Use Euler's Method to approximate the definite integral
#int_{1}^2(4x^2)dx# .For simplicity, let us use the step size
#Deltax=1# .Let
#I(t)=int_{1}^t(4x^2)dx# .So, we wish to approximate
#I(2)=int_{1}^2(4x^2)dx# Note that by Fundamental Theorem of Calculus I,
#I'(t)=4t^2# Now, let us start approximating.
#I(1)=\int_{1}^{1}(4x^2)dx=0# By linear approximation,
#I(0) approx I(1)+I'(1)cdot Delta x=0+3cdot1=3# #I(1) approx I(0)+I'(0)cdot Delta x approx3+4cdot1=7# #I(2) approx I(1)+I'(1)cdot Delta x approx 7+3cdot1=10# Hence,
#I(2)=int_{1}^2(4x^2)dx approx 10#
I hope that this was helpful.