8. Definite Integrals

{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.

Approximate the area under the curve $f\left(x\right)=\frac{2}{3x+5}$ over the interval $[4,10]$ using the Midpoint Riemann sum with $n=3$ subintervals.

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(c) Calculate the left and right Riemann sums for the given value of n.

∫₃⁶ (1―2𝓍) d𝓍 ; n = 6