{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture31 - Use the Fundamental Theorem of Calculus to...

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
§ 6.3 Definite Integrals and the Fundamental Theorem Math 121 Lecture 31 ! The Definite Integral ! Calculating Definite Integrals ! The Fundamental Theorem of Calculus ! Area Under a Curve as an Antiderivative
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
! x = ( b a )/ n , x 1 , x 2 , …., x n are selected points from a partition [ a , b ]. Calculate the following integral. The figure shows the graph of the function f ( x ) = x + 0.5. Since f ( x ) is nonnegative for 0 " x " 1, the definite integral of f ( x ) equals the area of the shaded region in the figure below. 1 0.5 1
Image of page 2
The region consists of a rectangle and a triangle. By geometry, Thus the area under the graph is 0.5 + 0.5 = 1, and hence
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Calculate the following integral. The figure shows the graph of the function f ( x ) = x on the interval -1 " x " 1. The area of the triangle above the x -axis is 0.5 and the area of the triangle below the x -axis is 0.5. Therefore, from geometry we find that
Image of page 4
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Use the Fundamental Theorem of Calculus to calculate the following integral. An antiderivative of 3 x 1/3 – 1 – e 0.5 x is . Therefore, by the fundamental theorem, ( Heat Diffusion ) Some food is placed in a freezer. After t hours the temperature of the food is dropping at the rate of r ( t ) degrees Fahrenheit per hour, where (a) Compute the area under the graph of y = r ( t ) over the interval 0 " t " 2. (b) What does the area in part (a) represent? (a) To compute the area under the graph of y = r ( t ) over the interval 0 " t " 2, we evaluate the following. (b) Since the area under a graph can represent the amount of change in a quantity , the area in part (a) represents the amount of change in the temperature between hour t = 0 and hour t = 2. That change is 24.533 degrees Fahrenheit....
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern