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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Evaluate the integral, if it exists.

0 2 y 2 1 + y 3   d y

To determine

To evaluate:

02y21+y3dy if it exists

Explanation

1) Concept:

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

2) Theorem and Rule:

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:

xndx=xn+1n+1+C

3) Given:

02y21+y3dy

4) Calculation:

Consider, 02y21+y3dy

Since the function is continuous in the given interval, the integral exists.

By applying substitution rule for definite integrals

Let u=1+y3

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