Answered: b+c f(x + c) dx f(u) du. la+c | bartleby

Calculus: Early Transcendentals

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Perhaps the simplest change of variables is the
shift or translation given by u = x + c, where c is a real number.
a. Prove that shifting a function does not change the net area under
the curve, in the sense that
b. Draw a picture to illustrate this change of variables in the case
where ƒ(x) = sin x, a = 0, b = p, and c = ∏/2.

b+c
f(x + c) dx
f(u) du.
la+c

Expert Solution

Step 1

(a) Prove:

$\int_{a}^{b} f (x + c) d x = \int_{a + c}^{b + c} f (u) d u$

Proof:

Consider Left Hand side:

$\int_{a}^{b} f (x + c) d x$ .

Let $x + c = u \dots (1)$

Implies $d x = d u$ .

Use (1) to find the change in limits

$a \to a + c$

$b \to b + c$

Substitute $x + c = u$ , $d x = d u$ into $\int_{a}^{b} f (x + c) d x$ and change the limits to get the new integral.

$\int_{a + c}^{b + c} f (u) d u$ = Right Hand Side.

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