91Ó°ÊÓ

Compute Midpoint, Trapezoidal and Simpson's Rule approximations by hand (leave your answer as a fraction) for \(n=4\). $$\int_{0}^{1}\left(x^{2}+1\right) d x$$

List the evaluation points corresponding to the midpoint of each sub interval, sketch the function and approximating rectangles and evaluate the Riemann sum. $$f(x)=x^{2}+1, \quad \text { (a) }[0,1], n=4 ; \quad \text { (b) }[0,2], n=4$$

A calculation is described in words. Translate each into summation notation and then compute the sum. The sum of the squares of the first 50 positive integers.

In exercises \(1-4,\) sketch several members of the family of functions defined by the antiderivative. $$\int x^{3} d x$$

Use the given substitution to evaluate the indicated integral. $$\int x^{2} \sqrt{x^{3}+2} d x, u=x^{3}+2$$

Use the Midpoint Rule to estimate the value of the integral (obtain two digits of accuracy). $$\int_{0}^{3}\left(x^{3}+x\right) d x$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{2}(2 x-3) d x$$

List the evaluation points corresponding to the midpoint of each sub interval, sketch the function and approximating rectangles and evaluate the Riemann sum. $$f(x)=x^{3}-1, \quad \text { (a) }[1,2], n=4 ; \quad \text { (b) }[1,3], n=4$$

Use Part I of the Fundamental Theorem to compute each integral exactly. $$\int_{0}^{3}\left(x^{2}-2\right) d x$$

Use the given substitution to evaluate the indicated integral. $$\int x^{3}\left(x^{4}+1\right)^{-2 / 3} d x, u=x^{4}+1$$