Rienmann sums

Question 1

Estimate the area under f(x) = x^2 on [1, 3] using a Left Riemann Sum with n = 2.

Answer 1

Delta x = 1. Sum = f(1)(1) + f(2)(1) = 1 + 4 = 5.

Question 2

Estimate the area under f(x) = x^2 on [1, 3] using a Right Riemann Sum with n = 2.

Answer 2

Delta x = 1. Sum = f(2)(1) + f(3)(1) = 4 + 9 = 13.

Question 3

For f(x) = 2x + 1 on [0, 4] with n = 4, calculate the Left Riemann Sum.

Answer 3

Delta x = 1. Sum = 1 + 3 + 5 + 7 = 16.

Question 4

For f(x) = 2x + 1 on [0, 4] with n = 4, calculate the Right Riemann Sum.

Answer 4

Delta x = 1. Sum = 3 + 5 + 7 + 9 = 24.

Question 5

Table: t={0,2,4,6}, v(t)={10,15,20,25}. Estimate distance on [0,6] using a Left Sum.

Answer 5

Delta t = 2. Sum = 2(10 + 15 + 20) = 90.

Question 6

Table: t={0,2,4,6}, v(t)={10,15,20,25}. Estimate distance on [0,6] using a Right Sum.

Answer 6

Delta t = 2. Sum = 2(15 + 20 + 25) = 120.

Question 7

Estimate the area under f(x) = 1/x on [1, 2] using a Right Riemann Sum with n = 2.

Answer 7

Delta x = 0.5. Sum = 0.5(2/3 + 1/2) = 7/12 or 0.583.

Question 8

Calculate the Left Riemann Sum for f(x) = sin(x) on [0, pi] with n = 2.

Answer 8

Delta x = pi/2. Sum = (pi/2)(0 + 1) = pi/2.

Question 9

Calculate the Right Riemann Sum for f(x) = sin(x) on [0, pi] with n = 2.

Answer 9

Delta x = pi/2. Sum = (pi/2)(1 + 0) = pi/2.

Question 10

Find the area of f(x) = 5 on the interval [2, 10] using a Right Riemann Sum with n = 4.

Answer 10

Delta x = 2. Sum = 2(5 + 5 + 5 + 5) = 40.