Evaluate the following summation:
Table of contents
- 0. Functions4h 53m
- 1. Limits and Continuity2h 2m
- 2. Intro to Derivatives1h 33m
- 3. Techniques of Differentiation2h 18m
- 4. Derivatives of Exponential & Logarithmic Functions1h 16m
- 5. Applications of Derivatives2h 19m
- 6. Graphical Applications of Derivatives6h 0m
- 7. Antiderivatives & Indefinite Integrals48m
- 8. Definite Integrals4h 36m
- 9. Graphical Applications of Integrals1h 43m
- 10. Integrals of Inverse, Exponential, & Logarithmic Functions21m
- 11. Techniques of Integration2h 7m
- 12. Trigonometric Functions6h 54m
- Angles29m
- Trigonometric Functions on Right Triangles1h 8m
- Solving Right Triangles23m
- Trigonometric Functions on the Unit Circle1h 19m
- Graphs of Sine & Cosine46m
- Graphs of Other Trigonometric Functions32m
- Trigonometric Identities52m
- Derivatives of Trig Functions42m
- Integrals of Basic Trig Functions28m
- Integrals of Other Trig Functions10m
- 13: Intro to Differential Equations2h 23m
- 14. Sequences & Series2h 8m
- 15. Power Series2h 19m
- 16. Probability & Calculus45m
8. Definite Integrals
Riemann Sums
Multiple Choice
For the following graph, write a Reimann sum using left endpoints to approximate the area under the curve over [0,6] with 6 subintervals.

A
9.62
B
6.62
C
7.16
D
8.15
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Verified step by step guidance1
Step 1: Understand the problem. We are tasked with approximating the area under the curve f(x) = 3 - √x over the interval [0,6] using a Riemann sum with 6 subintervals and left endpoints.
Step 2: Divide the interval [0,6] into 6 equal subintervals. The width of each subinterval (Δx) is calculated as Δx = (6 - 0)/6 = 1.
Step 3: Identify the left endpoints of each subinterval. The left endpoints are x = 0, 1, 2, 3, 4, and 5.
Step 4: Evaluate the function f(x) = 3 - √x at each left endpoint. This gives f(0), f(1), f(2), f(3), f(4), and f(5).
Step 5: Multiply each function value by the width of the subinterval (Δx = 1) and sum them up to approximate the area under the curve. The Riemann sum is Σ [f(x_i) * Δx] for i = 0 to 5.
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