How do you find the area using the trapezoidal approximation method, given #e^(x^2)#, on the interval [0,1] with n=10?

Question

Emma Aldridge · Accepted Answer

# int_0^1 e^(x^2) \ dx ~~ 1.4672 # (4dp)
The values of #f(x)=e^(x^2)# are tabulated as follows (using Excel) working to 6dp.

The Trapezium Rule:

# int_a^bydx ~~ h/2{(y_0+y_n)+2(y_1+y_2+...+y_(n-1))}#

uses a series of two consecutive ordinates and a best fit straight line to form trapeziums to approximate the area under a curve, It will have 100% accuracy if #y=f(x)# is a straight line and typically provides a good estimate provided #n# is chosen appropriately.

So,

# int_0^1 e^(x^2) \ dx ~~ 0.1/2 { (1 + 2.718281) + #
# " " 2(1.01005 + 1.04081 + 1.094174 + 1.17351 + 1.284025 +#
# " " 1.433329 + 1.632316 + 1.89648 + 2.247907) } #
# " "= 0.05 { + 3.718281 + 2(12.812606) } #
# " "= 0.05 { + 3.718281 + 25.625212 }#
# " "= 0.05 { + 29.343493 }#
# " "= 1.467174#

Elijah Abram · Answer

undefined To find the area using the trapezoidal approximation method for the function $ e^{x^2} $ on the interval $[0, 1]$ with $ n = 10 $, follow these steps:

1. Divide the interval $[0, 1]$ into $ n $ subintervals. Since $ n = 10 $, each subinterval has a width of $ \Delta x = \frac{1 - 0}{10} = 0.1 $.

2. Compute the function values at the endpoints of each subinterval and the midpoints of the subintervals. For $ n = 10 $, you will have 11 points: $ x_0 = 0, x_1 = 0.1, x_2 = 0.2, \ldots, x_{10} = 1 $.

3. Using the trapezoidal rule formula, calculate the area of each trapezoid formed by adjacent points. The formula is:
   $$
   \text{Area of trapezoid} = \frac{1}{2} \left[ f(x_i) + f(x_{i+1}) \right] \Delta x
   $$
   where $ x_i $ and $ x_{i+1} $ are the endpoints of the subinterval and $ f(x_i) $ and $ f(x_{i+1}) $ are the function values at those points.

4. Sum up the areas of all the trapezoids to get the total approximate area under the curve.

In this case, with $ n = 10 $, you would calculate the area using the trapezoidal rule with the 11 points obtained from dividing the interval $[0, 1]$ into 10 subintervals.