Math Problem Statement

Find the approximations T10 and M10 for 2

1 17 e1⁄x dx. (Round your answers to six decimal places.)

Solution

You are being asked to find two approximations, T10T_{10} (the Trapezoidal Rule with n=10n = 10) and M10M_{10} (the Midpoint Rule with n=10n = 10) for the definite integral:

I=217e1/xdx.I = \int_2^{17} e^{1/x} \, dx.

Step 1: Trapezoidal Rule Approximation (T10T_{10})

The Trapezoidal Rule is given by the formula: Tn=ba2n[f(a)+2i=1n1f(xi)+f(b)],T_n = \frac{b - a}{2n} \left[ f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right], where a=2a = 2, b=17b = 17, n=10n = 10, and xi=a+ihx_i = a + i \cdot h, with h=banh = \frac{b - a}{n}.

  1. Compute h=17210=1.5h = \frac{17 - 2}{10} = 1.5.
  2. The values of xix_i for i=1,2,,9i = 1, 2, \dots, 9 are 2+1.5i2 + 1.5i, or equivalently: x1=3.5,x2=5.0,x3=6.5,,x9=15.5.x_1 = 3.5, \, x_2 = 5.0, \, x_3 = 6.5, \dots, x_9 = 15.5.
  3. The trapezoidal approximation becomes: T10=1.52[f(2)+2(f(3.5)+f(5)++f(15.5))+f(17)],T_{10} = \frac{1.5}{2} \left[ f(2) + 2 \left( f(3.5) + f(5) + \cdots + f(15.5) \right) + f(17) \right], where f(x)=e1/xf(x) = e^{1/x}.

Step 2: Midpoint Rule Approximation (M10M_{10})

The Midpoint Rule is given by the formula: Mn=bani=1nf(xi1+xi2).M_n = \frac{b - a}{n} \sum_{i=1}^{n} f\left(\frac{x_{i-1} + x_i}{2}\right).

  1. Again, with a=2a = 2, b=17b = 17, and n=10n = 10, we have h=1.5h = 1.5.
  2. The midpoints are the values xi1+xi2\frac{x_{i-1} + x_i}{2}, which for i=1,2,,10i = 1, 2, \dots, 10, become: Midpoints=2.75,4.25,5.75,7.25,8.75,10.25,11.75,13.25,14.75,16.25.\text{Midpoints} = 2.75, 4.25, 5.75, 7.25, 8.75, 10.25, 11.75, 13.25, 14.75, 16.25.
  3. The midpoint approximation becomes: M10=1.5(f(2.75)+f(4.25)+f(5.75)++f(16.25)),M_{10} = 1.5 \left( f(2.75) + f(4.25) + f(5.75) + \cdots + f(16.25) \right), where f(x)=e1/xf(x) = e^{1/x}.

Step 3: Numerical Calculation

Now, we substitute the function values for f(x)=e1/xf(x) = e^{1/x} into the formulas for both T10T_{10} and M10M_{10}.

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Math Problem Analysis

Mathematical Concepts

Numerical Integration
Trapezoidal Rule
Midpoint Rule
Definite Integral

Formulas

Trapezoidal Rule: T_n = (b - a) / (2n) [f(a) + 2 * Σ f(x_i) + f(b)]
Midpoint Rule: M_n = (b - a) / n * Σ f((x_(i-1) + x_i) / 2)

Theorems

Numerical approximation for definite integrals

Suitable Grade Level

College Calculus or Advanced High School Calculus