Question

work through the following steps to evaluate $\int_{0}^{7}(x^{2}+4)dx$.
a) we know that $a = \square$ and $b = \square$.
b) using $n$ subintervals, $\delta x=\square$
c) assume that the sample points in each interval are right endpoints. find the following sample points:
$x_{1}=\square$
$x_{2}=\square$
$x_{3}=\square$
in general, the $i$th sample point is $x_{i}=\square$ note: your answer will be an expression in terms of $i$ and $n$.

Explanation:

Step1: Determine \(a\) and \(b\)

For the definite integral \(\int_{a}^{b}f(x)dx=\int_{0}^{7}(x^{2}+4)dx\), by comparing, we get \(a = 0\) and \(b=7\).

Step2: Calculate \(\Delta x\)

The formula for \(\Delta x=\frac{b - a}{n}\). Substituting \(a = 0\) and \(b = 7\) into the formula, we have \(\Delta x=\frac{7-0}{n}=\frac{7}{n}\).

Step3: Find sample points \(x_1,x_2,x_3,x_i\)

Since \(x_i=a + i\Delta x\) (right - endpoint formula, \(a = 0\) and \(\Delta x=\frac{7}{n}\)):

• For \(i = 1\), \(x_1=0+1\times\frac{7}{n}=\frac{7}{n}\)
• For \(i = 2\), \(x_2=0 + 2\times\frac{7}{n}=\frac{14}{n}\)
• For \(i = 3\), \(x_3=0+3\times\frac{7}{n}=\frac{21}{n}\)
• In general, \(x_i=0 + i\times\frac{7}{n}=\frac{7i}{n}\)

Answer:

a) \(a = 0\), \(b = 7\)
b) \(\Delta x=\frac{7}{n}\)
c) \(x_1=\frac{7}{n}\), \(x_2=\frac{14}{n}\), \(x_3=\frac{21}{n}\), \(x_i=\frac{7i}{n}\)