14.6.1 Approximate calculation of integrals

Problem: Calculate the integral \( \int_{1}^{2} f(x) d x \) with the step \( h=0,25 \), using the formula: a) of central rectangles; estimate the error by the Runge rule and a priori error estimate; б) of trapezoids; estimate the error by the Runge rule and the priori error estimate; clarify the result by Runge; в) Simpson's; estimate the error by the Runge. Intermediate calculations are carried out with six significant figures. Calculate the arguments of trigonometric functions in radians. Take into account the error while writing the answers. \[ \int_{1}^{2} f(x) d x-\text { ?, where } f(x)=e^{x \sqrt{x}} \text {, step } h=0,25 \text {. } \]