8. Definite Integrals

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

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

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral.

∫₃⁶ (1―2𝓍) d𝓍 ; n = 6

{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.

∫₀² (𝓍²―2) d𝓍 ; n = 4

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

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral..

∫₀² (𝓍²―2) d𝓍 ; n = 4