(Solved): THE PRISMOIDAL FORMULA Using the Trapezoidal Rule to Estimate the Volume Barrels and Flower Vases e ...

THE PRISMOIDAL FORMULA Using the Trapezoidal Rule to Estimate the Volume Barrels and Flower Vases etc. 1. Suppose a solid of revolution is bounded by two planes a distance $$h$$ units apart which are perpendicular to the axis of rotation. If the solid were to be oriented with its axis of rotation parallel to the $$x$$-axis and centered so that the two planes intersect the $$x$$-axis at $$?\frac{h}{2}$$ and $$\frac{h}{2}$$ then the outline of the solid is given by $$y=f(x)$$. a) Sketch the solid oriented and centered as described. Include as much detail as possible. Set up the integral to compute the volume of the solid (use the washer method). Write the approximate volume using the trapezoid rule with two partitions to evaluate the integral. Then substitute the three quantities below to get you Prismoidal Formula. $$A_t=$$ area of left-most face, $$A_M=$$ area of middle cross section, $$A_n=$$ area of right-most face b) Use your formula above to approximate the volume of a cone frustum shown. How accurate is this approximation? Hint: Use the error bound below. If this volume is written with integrand $$f(x)$$ then its second derivative will be a constant. That constant corresponds to the max value needed in the formula. $$?E??\frac{(b?a{)}^3}{12n^2}\left[\max ?f^{??}(x)?\right],a<x<b$$