8. Definite Integrals

Express the integral ${\int }_0^{2}3x^2+1dx$ as a limit of Riemann sums using right endpoints. Do not evaluate the limit.

{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.

(a) Write the midpoint Riemann sum in sigma notation for an arbitrary value of n.

∫₀⁴ (4𝓍― 𝓍²) d𝓍

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.

ƒ(𝓍) = tan⁻¹ (3x - 1) on [0,1]

(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

Evaluate the following summation: 

${\sum }_{i=1}^4{i}^2-3i+8$

Evaluate the following summation: 

${\sum }_{k=1}^4{\left(\frac{k}{{}_{}2}\right)}^2$