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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1:14 minutes

Problem 5.R.102a

Textbook Question

Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.
(a) Evaluate H (0) .

Verified step by step guidance

Step 1: Understand the problem. The function H(𝓍) is defined as an integral from 0 to 𝓍 of √(4 − t²) dt. For part (a), we are tasked with evaluating H(0), which means substituting 𝓍 = 0 into the integral.

Step 2: Substitute 𝓍 = 0 into the integral definition. This gives H(0) = ∫₀⁰ √(4 − t²) dt.

Step 3: Recognize that the integral's limits are both 0. When the upper and lower limits of an integral are the same, the integral evaluates to 0. This is because there is no interval over which to integrate.

Step 4: Conclude that H(0) = 0 based on the property of definite integrals where the limits are equal.

Step 5: Reflect on the result. This step demonstrates the importance of understanding integral properties and how they simplify calculations in specific cases.

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Key Concepts

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

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. In this case, H(𝓍) is defined as the integral of √(4 - t²) from 0 to 𝓍. Evaluating a definite integral involves finding the antiderivative of the integrand and applying the Fundamental Theorem of Calculus.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then the definite integral of f from a to b is F(b) - F(a). This theorem allows us to evaluate H(𝓍) by finding an antiderivative of the integrand and substituting the limits of integration.

Geometric Interpretation of Integrals

Integrals can be interpreted geometrically as the area under a curve. For the function √(4 - t²), which describes a semicircle, the integral from 0 to 𝓍 gives the area of the corresponding segment of the semicircle. Understanding this geometric interpretation aids in visualizing the result of the integral and its evaluation at specific points like H(0).