Question

find the trapezoidal riemann sum approximation of the integral from 1 to 5 of the square root of quantity x squared plus 3 close quantity times dx using four equal partitions.

Answer 1

14.115

Explanation:

Recall Trapezoidal Rule

`\displaystyle \int\limits^b_a {f(x)} \, dx=(1)/(2)\biggr((b-a)/(n)\biggr)\biggr[f(x_0)+2\bigr(f(x_1)+f(x_2)+\text{... }+f(x_(n-1))\bigr)+f(x_n)\biggr]`

Make necessary substitutions and evaluate

`\displaystyle \int\limits^5_1 {√(x^2+3)} \, dx=(1)/(2)\biggr((5-1)/(4)\biggr)\biggr[2+2\bigr(√(7)+√(12)+√(19)\bigr)+√(28)\biggr]\approx14.115`

Note that the actual value of the integral is
`\displaystyle \int\limits^5_1 {√(x^2+3)} \, dx\approx14.078`, so we can see how accurate the Trapezoidal rule is with just a few equal partions.