Question

Which expression is the summation notation that represents the integral ∫ 1

5
​
(x 2
+2x)dx ? A lim n→[infinity]
​
∑ k=1
n
​
( n
1
​
)((1+( n
1
​
k)) 2
+2(1+( n
1
​
k))) B lim n→[infinity]
​
∑ k=1
n
​
( n
5
​
)((1+( n
5
​
k)) 2
+2(1+( n
5
​
k))) C lim n→[infinity]
​
∑ k=1
n
​
( n
4
​
)((1+( n
4
​
k)) 2
+(1+( n
4
​
k))) D lim n→[infinity]
​
∑ k=1
n
​
( n
4
​
)((1+( n
4
​
k)) 2
+2(1+( n
4
​
k)))

Answer 1

The integral ∫ (x^2 + 2x)dx from 1 to 5 can be approximated using the right endpoint Riemann sum, which is represented by the summation notation as follows:

lim n→∞ ∑ (from k=1 to n) of [ (b-a)/n * f(a + k*((b-a)/n)) ]

For this question, we use a = 1 and b = 5. The function f(x) is x^2 + 2x. Substituting the given values, the equation as per our parameters becomes:

lim n→∞ ∑ (from k=1 to n) of [ (5-1)/n * ((1 + k*((5-1)/n))^2 + 2*(1 + k*((5-1)/n))) ]

This equation simplifies as follows:

lim n→∞ ∑ (from k=1 to n) of [ ( 1/n*(4) ) * ((1+(4/n)*k)^2 + 2*(1 + (4/n)*k)) ]

Looking at the provided options, it is clear that it matches option B:

lim n→∞ ∑ k=1 to n of ( 1/n*(4) ) * ((1+(4/n)*k)^2 + 2*(1 + (4/n)*k))

Hence, the correct answer is option B.