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Let
f be the function defined by
f(x)=x^(2). If four subintervals of equal length are used, what is the value of the right Riemann sum approximation for
int_(2)^(6)x^(2)dx ?
Round to the nearest thousandth if necessary.
Answer:

Let $f$ be the function defined by $f(x)=x^{2}$ . If four subintervals of equal length are used, what is the value of the right Riemann sum approximation for $\int_{2}^{6} x^{2} d x$ ? $\newline$ Round to the nearest thousandth if necessary. $\newline$ Answer:

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Find trigonometric ratios using reference angles

Full solution

Q. Let

f

be the function defined by

f(x)=x^{2}

. If four subintervals of equal length are used, what is the value of the right Riemann sum approximation for

\int_{2}^{6} x^{2} d x

?

\newline

Round to the nearest thousandth if necessary.

\newline

Answer:

Determine Width of Subintervals: Determine the width of each subinterval. Since we are integrating from $x = 2$ to $x = 6$ , the total interval length is $6 - 2 = 4$ . With four subintervals of equal length, each subinterval will have a width of $\frac{4}{4} = 1$ .
Identify Endpoints of Subintervals: Identify the endpoints of each subinterval. The subintervals are $[2, 3]$ , $[3, 4]$ , $[4, 5]$ , and $[5, 6]$ . Since we are using a right Riemann sum, we will evaluate the function at the right endpoints of each subinterval, which are $x = 3$ , $x = 4$ , $x = 5$ , and $x = 6$ .
Evaluate Function at Endpoints: Evaluate the function at each right endpoint. $\newline$ $f(3) = 3^2 = 9$ $\newline$ $f(4) = 4^2 = 16$ $\newline$ $f(5) = 5^2 = 25$ $\newline$ $f(6) = 6^2 = 36$
Calculate Rectangle Areas: Multiply each function value by the width of the subintervals to find the area of each rectangle. Since the width of each subinterval is $1$ , the area of each rectangle is simply the function value at the right endpoint. Area of rectangle $1$ : $1 \times f(3) = 1 \times 9 = 9$ Area of rectangle $2$ : $1 \times f(4) = 1 \times 16 = 16$ Area of rectangle $3$ : $1 \times f(5) = 1 \times 25 = 25$ Area of rectangle $4$ : $1 \times f(6) = 1 \times 36 = 36$
Sum Areas for Riemann Sum: Sum the areas of the rectangles to find the right Riemann sum approximation. $\newline$ Right Riemann sum $= 9 + 16 + 25 + 36 = 86$

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