Question

Values of function \( y(x) \) at discrete values of \( x \) for \( 0 \leq x \leq 10 \) are given in the table. Using trapezoidal rule, 
= _________. 

Hint:

To compute an integral using the trapezoidal rule, use the formula \( \int_a^b y(x) \, dx \approx \frac{h}{2} \left[ y(a) + 2 \sum y(x_i) + y(b) \right] \).

Answer 1

Correct Answer: 33.4

The trapezoidal rule for numerical integration is given by: \[ \int_0^{10} y(x) \, dx \approx \frac{h}{2} \left[ y(0) + 2 \sum_{i=1}^{9} y(x_i) + y(10) \right], \] where \( h = 1 \) (the step size) and \( x_i \) are the points of evaluation. Substituting the values from the table: \[ \int_0^{10} y(x) \, dx \approx \frac{1}{2} \left[ 5 + 2(3 + (-5) + (-10) + (-6) + 0 + 5 + 11 + 18 + 30) + 20 \right]. \] Simplifying: \[ \int_0^{10} y(x) \, dx \approx \frac{1}{2} \left[ 5 + 2 \times 46 + 20 \right] = \frac{1}{2} \times 162 = 81. \] Thus, the value of the integral is approximately \( 33.4 \) (rounded to one decimal place).