Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

8. Definite Integrals

Introduction to Definite Integrals

Calculus

8. Definite Integrals

Introduction to Definite Integrals

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Multiple Choice

Which of the following limits is equal to the definite integral from $2$ to $5$ of $1 + x^{4}$ $d x$ ?

lim_{n \to \infty} \sum i = 1 n [1 + (2 + \frac{3 i}{n}) (4)] \cdot \frac{3}{n}

lim_{n \to \infty} \sum i = 1 n [1 + (2 + \frac{i}{n}) (4)] \cdot \frac{1}{n}

lim_{n \to \infty} \sum i = 1 n [1 + (2 + \frac{5 i}{n}) (4)] \cdot \frac{5}{n}

lim_{n \to \infty} \sum i = 1 n [1 + (5 + \frac{3 i}{n}) (4)] \cdot \frac{3}{n}

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Verified step by step guidance

Step 1: Recall the definition of a definite integral as the limit of a Riemann sum. The definite integral from a to b of a function f(x) dx can be expressed as lim_{n \to \infty} \sum_{i=1}^n f(x_i) \cdot \Delta x, where \Delta x = \frac{b-a}{n} and x_i represents the sample points within each subinterval.

Step 2: Identify the interval of integration. In this problem, the definite integral is from 2 to 5, so the interval is [2, 5]. The width of each subinterval is \Delta x = \frac{5-2}{n} = \frac{3}{n}.

Step 3: Determine the sample points x_i within the interval. A common choice for sample points is the left endpoint of each subinterval, which can be expressed as x_i = 2 + i \cdot \Delta x = 2 + \frac{3i}{n}.

Step 4: Substitute the function f(x) = 1 + x^4 into the Riemann sum formula. The sum becomes \sum_{i=1}^n \left[1 + \left(2 + \frac{3i}{n}\right)^4\right] \cdot \Delta x, where \Delta x = \frac{3}{n}.

Step 5: Compare the given options to the derived Riemann sum expression. The correct limit is lim_{n \to \infty} \sum_{i=1}^n \left[1 + \left(2 + \frac{3i}{n}\right)^4\right] \cdot \frac{3}{n}, as it matches the interval, sample points, and function provided in the problem.