Evaluate the integral. ∫−11(x−3)2dx

Question

Evaluate the integral.  ∫−11(x−3)2dx

Fesion · Accepted Answer

Step 1  Given :  ∫−11(x−3)2dx  To find the value of the above integral.  Step 2  ∫−11(x−3)2dx  =∫−11(x2−6x+9)dx   (since (a−b)2=a2−2ab+b2)  =[x33−6x22+9x]−11  =(133−6122+9(1))−((−1)33−6(−1)22+9(−1))  =(13−3+9)−(−13−3−9)  =(13+6)−(−13−12)  =(1+183)−(−1−363)=193+373=563  Thus, ∫−11(x−3)2dx=563.

Geraldine Flores · Accepted Answer

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}

Evaluate the integral. ∫−11(x−3)2dx

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