Step by step solution to "Calculate the integral. ∫06|3−x|dx"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

David Troyer

Answered question

2021-12-30

Calculate the integral.

\int_{0}^{6} | 3 - x | d x

Answer & Explanation

Archie Jones

Beginner2021-12-31Added 34 answers

Step 1
Given:

I = \int_{0}^{6} | 3 - x | d x

for evaluating given integral, we break in two parts and integrate it
Step 2
so,

I = \int_{0}^{6} | 3 - x | d x

= \int_{0}^{3} (3 - x) d x + \int_{3}^{6} (x - 3) d x

(∵ \int (x - a) d x = \frac{{(x - a)}^{2}}{2} + c)

= - \frac{1}{2} {[{(x - 3)}^{2}]}_{0}^{3} + \frac{1}{2} {[{(x - 3)}^{2}]}_{3}^{6}

= - \frac{1}{2} [{(3 - 3)}^{2} - {(0 - 3)}^{2}] + \frac{1}{2} [{(6 - 3)}^{2} - {(3 - 3)}^{2}]

= - \frac{1}{2} [0 - 9] + \frac{1}{2} [9 - 0]

= \frac{9}{2} + \frac{9}{2}

=9
hence, given integral is equal to 9.

Ana Robertson

Beginner2022-01-01Added 26 answers

Step 1
Given:

\int_{0}^{6} | 3 - x | d x

Step 2
Solution

\int_{0}^{3} 3 - x d x + \int_{3}^{6} - (3 - x) d x

Evaluate the integrals

\frac{9}{2} + \frac{9}{2}

Calculate the sum
Step 3
Answer:
9

Vasquez

Expert2022-01-07Added 669 answers

$\int (3 - x) d x$
$\int (3 - x) d x = - \frac{x^{2}}{2} + 3 x$
Let's calculate a definite integral:
$\int_{0}^{6} 3 - x d x = (- \frac{x^{2}}{2} + 3 x) |_{0}^{6}$
F(6)=0
F(0)=0
I=6-(0)=6
Answer:
6

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