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: 

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

Evaluate the following summation: 

${\sum }_{i=0}^2\frac{2i}{3}$

Evaluate the following summation (make sure your calculator is in radian mode): 

${\sum }_{i=1}^4\sin \left(\frac{\pi i}{8}\right)$

Evaluate the following summation: 

${\sum }_{t=2}^{6}5t+\frac{4}{t}$

Evaluate the following summation: 

${\sum }_{i=0}^{4}2i+1$

Evaluate the following summation: 

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

Evaluate the following summation (make sure your solution is in radians):

${\sum }_{k=0}^{5}2\tan \left(\frac{\pi k}{3}\right)-\pi k$

For the following graph, write a Reimann sum using left endpoints to approximate the area under the curve over [0,5] with 5 subintervals.

For the following graph, write a Reimann sum using left endpoints to approximate the area under the curve over [0,6] with 6 subintervals.

Write the Riemann sum that would approximate the area of the following graph over the interval [0,3] using 3 subintervals.

Approximate the area under the curve $f\left(x\right)=\sqrt{x+3}$ over the interval $[1,5]$ using the Right Riemann sum with 8 subintervals.