AP Calculus AB 67 The Definite Integral of sinx from 0 to pi

>> YOUR LINK HERE: ___ http://youtube.com/watch?v=lm6GbNyXOYE

Please subscribe! / nickperich • AP Calculus AB 6.7: The Definite Integral of \\( \\sin(x) \\) from \\( 0 \\) to \\( \\pi \\) • #### Overview • In this section, students learn to evaluate the definite integral of the sine function, specifically from \\( 0 \\) to \\( \\pi \\). This integral is not only foundational in calculus but also has significant implications in various fields, including physics and engineering. • #### Definite Integral Setup • 1. **Expression of the Integral**: • The definite integral of the sine function over the interval from \\( 0 \\) to \\( \\pi \\) is expressed as: • \\[ • \\int_{0}^{\\pi} \\sin(x) \\, dx • \\] • 2. **Finding the Antiderivative**: • The antiderivative of \\( \\sin(x) \\) is: • \\[ • F(x) = -\\cos(x) + C • \\] • This is a crucial step as it allows us to evaluate the integral. • #### Evaluation Process • 3. **Applying the Fundamental Theorem of Calculus**: • To evaluate the definite integral, we apply the Fundamental Theorem of Calculus, which states: • \\[ • \\int_{a}^{b} f(x) \\, dx = F(b) - F(a) • \\] • In this case: • \\[ • \\int_{0}^{\\pi} \\sin(x) \\, dx = F(\\pi) - F(0) • \\] • 4. **Substituting the Limits**: • Substituting the upper and lower limits into the antiderivative: • \\[ • = \\left[-\\cos(\\pi)\ ight] - \\left[-\\cos(0)\ ight] • \\] • 5. **Simplifying the Expression**: • Evaluating the cosine values gives: • \\( \\cos(\\pi) = -1 \\) • \\( \\cos(0) = 1 \\) • Thus, the integral simplifies to: • \\[ • = -(-1) - (-1) = 1 + 1 = 2 • \\] • #### Geometric Interpretation • The definite integral represents the area under the curve of \\( \\sin(x) \\) from \\( x = 0 \\) to \\( x = \\pi \\). The sine function starts at \\( 0 \\), reaches its maximum value of \\( 1 \\) at \\( x = \\frac{\\pi}{2} \\), and returns to \\( 0 \\) at \\( x = \\pi \\). • The area calculated by the definite integral corresponds to the total positive area between the sine curve and the x-axis within the specified interval. This area can be visualized as a wave pattern, which has practical applications in analyzing oscillations, wave functions, and other periodic phenomena. • #### Importance • Understanding the definite integral of \\( \\sin(x) \\) from \\( 0 \\) to \\( \\pi \\) emphasizes the relationship between calculus and geometry, specifically in calculating areas under curves. This integral is an excellent example of the applications of the Fundamental Theorem of Calculus, providing insights into the behavior of trigonometric functions. • Students gain skills in integrating basic functions and interpreting results in a real-world context, laying the groundwork for more complex calculus concepts and applications in physics, engineering, and other scientific fields. • I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out: • / nickperich • Nick Perich • Norristown Area High School • Norristown Area School District • Norristown, Pa • #math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study

#############################

New on site