"Evaluate the integral. ∫−11(x−3)2dx" solved and explained

Ashley Searcy

Answered question

2021-11-18

Evaluate the integral.

\int_{- 1}^{1} {(x - 3)}^{2} d x

Fesion

Beginner2021-11-19Added 24 answers

Step 1
Given :

\int_{- 1}^{1} {(x - 3)}^{2} d x

To find the value of the above integral.
Step 2

\int_{- 1}^{1} {(x - 3)}^{2} d x

= \int_{- 1}^{1} (x^{2} - 6 x + 9) d x

(since

{(a - b)}^{2} = a^{2} - 2 a b + b^{2}

)

= {[\frac{x^{3}}{3} - 6 \frac{x^{2}}{2} + 9 x]}_{- 1}^{1}

= (\frac{1^{3}}{3} - 6 \frac{1^{2}}{2} + 9 (1)) - (\frac{{(- 1)}^{3}}{3} - 6 \frac{{(- 1)}^{2}}{2} + 9 (- 1))

= (\frac{1}{3} - 3 + 9) - (\frac{- 1}{3} - 3 - 9)

= (\frac{1}{3} + 6) - (\frac{- 1}{3} - 12)

= (\frac{1 + 18}{3}) - (\frac{- 1 - 36}{3}) = \frac{19}{3} + \frac{37}{3} = \frac{56}{3}

Thus,

\int_{- 1}^{1} {(x - 3)}^{2} d x = \frac{56}{3}

Geraldine Flores

Beginner2021-11-20Added 21 answers

Step 1: If f(x) is a continuous function from a to b, and if F(x) is its integral, then:

\int_{a}^{b} f (x) d x = F (x) ∣_{a}^{b} = F (b) - F (a)

Step 2: In this case,

f (x) = {(x - 3)}^{2}

. Find its integral.

\frac{x^{3}}{3} - 3 x^{2} + 9 x ∣_{- 1}^{1}

Step 3: Since

F (x) ∣_{a}^{b} = F (b) - F (a)

, expand the above into F(1)−F(−1):

(\frac{1^{3}}{3} - 3 \times 1^{2} + 9 \times 1) - (\frac{{(- 1)}^{3}}{3} - 3 {(- 1)}^{2} + 9 \times - 1)

Step 4: Simplify.

\frac{56}{3}

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