Let $f (x) = 4 x + 7, x_{1} = 2, x_{2} = 4, x_{3} = 6, x_{4} = 8$ , and $Δ x = 2$
(a) Find $\sum_{i = 1}^{4} f (x_{i}) Δ x$ .
(b) The sum in part (a) approximates a definite integral using rectangles. The height of each rectangle is given by the value of the function at the left endpoint. Write the definite integral that the sum approximates.

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$216$ is the final answer

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Step 1 ：Given the function $f (x) = 4 x + 7$ , the values $x_{1} = 2, x_{2} = 4, x_{3} = 6, x_{4} = 8$ , and $Δ x = 2$

Step 2 ：We need to find $\sum_{i = 1}^{4} f (x_{i}) Δ x$

Step 3 ：This involves substituting the values of $x_{1}, x_{2}, x_{3}, x_{4}$ into the function $f (x)$ , multiplying each result by $Δ x$ , and then summing them all up

Step 4 ：Substituting the values into the function gives $f (x_{1}) = 4 * 2 + 7 = 15$ , $f (x_{2}) = 4 * 4 + 7 = 23$ , $f (x_{3}) = 4 * 6 + 7 = 31$ , and $f (x_{4}) = 4 * 8 + 7 = 39$

Step 5 ：Multiplying each result by $Δ x$ gives $15 * 2 = 30$ , $23 * 2 = 46$ , $31 * 2 = 62$ , and $39 * 2 = 78$

Step 6 ：Summing these results gives $30 + 46 + 62 + 78 = 216$

Step 7 ：Thus, $\sum_{i = 1}^{4} f (x_{i}) Δ x = 216$

Step 8 ： $216$ is the final answer

Let f(x)=4x+7,x1=2,x2=4,x3=6,x4=8, and Δx=2(a) Find ∑i=14f(xi)Δx.(b) The sum in part (a) approximates a definite integral using rectangles. The height of each rectangle is given by the value of the function at the left endpoint. Write the definite integral that the sum approximates.

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