Evaluate the integral, if it exists. – 1 |dx

Answered: – 1 |dx

Math

Calculus

– 1 |dx

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section: Chapter Questions

Problem 32T

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Question

Evaluate the integral, if it exists.

With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.

Expert Solution

Step 1

Given: $I = \int_{0}^{4} |\sqrt{x} - 1| d x$

for evaluate given integral, we break it in two parts as given below

$\int_{0}^{4} |\sqrt{x} - 1| d x = \int_{0}^{1} |\sqrt{x} - 1| d x + \int_{1}^{4} |\sqrt{x} - 1| d x$

in first integral $\sqrt{x} < 1 \Rightarrow |\sqrt{x} - 1| = 1 - \sqrt{x}$

and

in second integral $\sqrt{x} > 1 \Rightarrow |\sqrt{x} - 1| = \sqrt{x} - 1$

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