Question

35 - 38. identifying definite integrals as limits of sums consider the following limits of riemann sums for a function f on a, b. identify f and express the limit as a definite integral.
35.

$$ \\lim _ { \\delta \ ightarrow 0 } \\sum _ { k = 1 } ^ { n } ( x _ { k } ^ { * 2 } + 1 ) \\delta x _ { k } $$

on 0, 2
36.

$$ \\lim _ { \\delta \ ightarrow 0 } \\sum _ { k = 1 } ^ { n } ( 4 - x _ { k } ^ { * 2 } ) \\delta x _ { k } $$

on -2, 2
37.

$$ \\lim _ { \\delta \ ightarrow 0 } \\sum _ { k = 1 } ^ { n } x _ { k } ^ { * } ( \\ln x _ { k } ^ { * } ) \\delta x _ { k } $$

on 1, 2

Explanation:

Step1: Recall the definition of definite integral

The definite integral of a function \(f(x)\) on the interval \([a,b]\) is defined as \(\int_{a}^{b}f(x)dx=\lim_{\Delta
ightarrow0}\sum_{k = 1}^{n}f(x_{k}^{*})\Delta x_{k}\), where \(x_{k}^{*}\) is a sample point in the \(k -\)th sub - interval \([x_{k-1},x_{k}]\) and \(\Delta x_{k}=x_{k}-x_{k - 1}\).

Step2: Identify the function \(f(x)\) for problem 35

For the limit \(\lim_{\Delta
ightarrow0}\sum_{k = 1}^{n}(x_{k}^{*2}+1)\Delta x_{k}\) on \([0,2]\), by comparing with the formula \(\lim_{\Delta
ightarrow0}\sum_{k = 1}^{n}f(x_{k}^{*})\Delta x_{k}\), we can see that \(f(x)=x^{2}+1\), \(a = 0\) and \(b=2\).
So, the definite integral is \(\int_{0}^{2}(x^{2}+1)dx\).

Step3: Identify the function \(f(x)\) for problem 36

For the limit \(\lim_{\Delta
ightarrow0}\sum_{k = 1}^{n}(4 - x_{k}^{*2})\Delta x_{k}\) on \([-2,2]\), comparing with \(\lim_{\Delta
ightarrow0}\sum_{k = 1}^{n}f(x_{k}^{*})\Delta x_{k}\), we have \(f(x)=4 - x^{2}\), \(a=-2\) and \(b = 2\).
So, the definite integral is \(\int_{-2}^{2}(4 - x^{2})dx\).

Step4: Identify the function \(f(x)\) for problem 37

For the limit \(\lim_{\Delta
ightarrow0}\sum_{k = 1}^{n}x_{k}^{}(\ln x_{k}^{})\Delta x_{k}\) on \([1,2]\), comparing with \(\lim_{\Delta
ightarrow0}\sum_{k = 1}^{n}f(x_{k}^{*})\Delta x_{k}\), we get \(f(x)=x\ln x\), \(a = 1\) and \(b=2\).
So, the definite integral is \(\int_{1}^{2}x\ln xdx\).

Answer:

• For problem 35: \(f(x)=x^{2}+1\), \(\int_{0}^{2}(x^{2}+1)dx\)
• For problem 36: \(f(x)=4 - x^{2}\), \(\int_{-2}^{2}(4 - x^{2})dx\)
• For problem 37: \(f(x)=x\ln x\), \(\int_{1}^{2}x\ln xdx\)