Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the area of the region bounded by the graphs of $f$ and $g$ is 1, then the area of the region bounded by the graphs of $h(x)=f(x)+C$ and $k(x)=g(x)+C$ is also $1 .$

Name: Problem 90, Chapter 7: Calculus
Uploaded: 2021-07-21T17:04:52-08:00
Duration: 1 min 45 s
Channel: Gregory Higby
Description: Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the area of the region bounded by the graphs of f and g is 1, then the area of the region bounded by the graphs of h(x)=f(x)+C and k(x)=g(x)+C is also 1 .

   Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If the area of the region bounded by the graphs of $f$ and $g$ is 1, then the area of the region bounded by the graphs of $h(x)=f(x)+C$ and $k(x)=g(x)+C$ is also $1 .$

Calculus

Ron Larson, Robert… 8th Edition

Chapter 7, Problem 90

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Solved by Expert Gregory Higby

07/21/2021

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We are given that the area of the region bounded by the graphs of f and g is 1. Show more…

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Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the area of the region bounded by the graphs of $f$ and $g$ is 1, then the area of the region bounded by the graphs of $h(x)=f(x)+C$ and $k(x)=g(x)+C$ is also $1 .$

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Key Concepts

Vertical Translation

Vertical translation refers to shifting a function’s graph upward or downward by adding (or subtracting) a constant to the function. Importantly, when the same constant is added to two functions, their vertical separation at any given point remains unchanged, because the constant cancels out when taking the difference between them.

Area Between Curves

The area between the graphs of two functions is typically determined by integrating the absolute difference of the functions over a specified interval. Since a vertical translation applied equally to both functions does not change the difference between them, the integral, and thus the area between the curves, remains unchanged.

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