   Chapter 4.R, Problem 20E

Chapter
Section
Textbook Problem

# Evaluate the integral, if it exists. ∫ 0 1 sin ( 3 π t )   d t

To determine

To evaluate:

01sin(3πt)dt if it exists

Explanation

1) Concept:

By using substitution rule for definite integrals and fundamental rule of calculus

2) Theorem:

Fundamental theorem of calculus:

If f is continuous on [a, b], then abfxdx=Fb-Fa.

Substitution rule for definite integrals:

If g' is continuous on [a, b] and f is continuous on the range of u=g(x) then

abfgxg'xdx=g(a)g(b)f(u)du

3) Formula:

sinxdx=-cosx+C

3) Given:

01sin(3πt)dt

4) Calculation:

Consider, 01sin(3πt)dt

Since sin(3πt) is continuous [0,1], the given integral exists.

By applying substitution rule for definite integrals,

Let u=3πt

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