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

Fundamental Theorem of Calculus

Calculus

8. Definite Integrals

Fundamental Theorem of Calculus

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2:55 minutes

Problem 5.3.59

Textbook Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫₁² (z² + 4) / z dz

Verified step by step guidance

Step 1: Simplify the integrand. The given integral is ∫₁² (z² + 4) / z dz. Split the fraction into two terms: (z² / z) + (4 / z). This simplifies to ∫₁² (z + 4/z) dz.

Step 2: Break the integral into two separate integrals. Using the linearity property of integrals, rewrite the expression as ∫₁² z dz + ∫₁² (4/z) dz.

Step 3: Compute the antiderivative of each term. For the first term, ∫ z dz, the antiderivative is (1/2)z². For the second term, ∫ (4/z) dz, the antiderivative is 4ln|z|.

Step 4: Apply the Fundamental Theorem of Calculus. Evaluate the antiderivatives at the upper limit (z = 2) and subtract the evaluation at the lower limit (z = 1). For (1/2)z², compute [(1/2)(2²) - (1/2)(1²)]. For 4ln|z|, compute [4ln(2) - 4ln(1)].

Step 5: Combine the results from both terms. Add the results from the evaluations of (1/2)z² and 4ln|z| to obtain the final value of the definite integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integrals

Definite integrals represent the signed area under a curve between two specified limits. They are calculated using the integral symbol with lower and upper bounds, indicating the interval over which the function is evaluated. The result of a definite integral is a numerical value that reflects the accumulation of quantities, such as area, over that interval.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with integration, providing a method to evaluate definite integrals. It states that if a function is continuous on an interval, the integral of its derivative over that interval equals the difference in the values of the original function at the endpoints. This theorem allows for the computation of definite integrals by finding an antiderivative of the integrand.

Antiderivatives

An antiderivative of a function is another function whose derivative yields the original function. Finding an antiderivative is essential for evaluating definite integrals using the Fundamental Theorem of Calculus. For example, if F(z) is an antiderivative of f(z), then the definite integral from a to b can be computed as F(b) - F(a), providing the net area under the curve of f(z) between those limits.