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Math  /  Calculus

QuestionApproximate ∫214x2dx\int_{2}^{14} x^{2} d x using each of the following Riemann sums with 4 subintervals of equal length. Do not simplify your answer. (a) Left Riemann Sum = □\square (b) Right Riemann Sum = □\square (c) Midpoint Rule == □\square

Studdy Solution

STEP 1

1. We are approximating the integral ∫214x2dx\int_{2}^{14} x^{2} dx.
2. We will use Riemann sums with 4 subintervals of equal length.
3. We will calculate the Left Riemann Sum, Right Riemann Sum, and Midpoint Rule.

STEP 2

1. Determine the width of each subinterval.
2. Calculate the Left Riemann Sum.
3. Calculate the Right Riemann Sum.
4. Calculate the Midpoint Rule.

STEP 3

Determine the width of each subinterval:
The interval from 22 to 1414 is divided into 4 subintervals, so the width Δx\Delta x is:
Δx=14−24=3\Delta x = \frac{14 - 2}{4} = 3

STEP 4

Calculate the Left Riemann Sum:
The left endpoints of the subintervals are x0=2x_0 = 2, x1=5x_1 = 5, x2=8x_2 = 8, x3=11x_3 = 11.
The Left Riemann Sum is:
Left Riemann Sum=Δx[f(x0)+f(x1)+f(x2)+f(x3)]\text{Left Riemann Sum} = \Delta x \left[ f(x_0) + f(x_1) + f(x_2) + f(x_3) \right]
Substitute the function values:
=3[(2)2+(5)2+(8)2+(11)2]= 3 \left[ (2)^2 + (5)^2 + (8)^2 + (11)^2 \right]

STEP 5

Calculate the Right Riemann Sum:
The right endpoints of the subintervals are x1=5x_1 = 5, x2=8x_2 = 8, x3=11x_3 = 11, x4=14x_4 = 14.
The Right Riemann Sum is:
Right Riemann Sum=Δx[f(x1)+f(x2)+f(x3)+f(x4)]\text{Right Riemann Sum} = \Delta x \left[ f(x_1) + f(x_2) + f(x_3) + f(x_4) \right]
Substitute the function values:
=3[(5)2+(8)2+(11)2+(14)2]= 3 \left[ (5)^2 + (8)^2 + (11)^2 + (14)^2 \right]

STEP 6

Calculate the Midpoint Rule:
The midpoints of the subintervals are x0=3.5x_0 = 3.5, x1=6.5x_1 = 6.5, x2=9.5x_2 = 9.5, x3=12.5x_3 = 12.5.
The Midpoint Rule is:
Midpoint Rule=Δx[f(x0)+f(x1)+f(x2)+f(x3)]\text{Midpoint Rule} = \Delta x \left[ f(x_0) + f(x_1) + f(x_2) + f(x_3) \right]
Substitute the function values:
=3[(3.5)2+(6.5)2+(9.5)2+(12.5)2]= 3 \left[ (3.5)^2 + (6.5)^2 + (9.5)^2 + (12.5)^2 \right]
The approximations are: (a) Left Riemann Sum = 3[(2)2+(5)2+(8)2+(11)2]3 \left[ (2)^2 + (5)^2 + (8)^2 + (11)^2 \right] (b) Right Riemann Sum = 3[(5)2+(8)2+(11)2+(14)2]3 \left[ (5)^2 + (8)^2 + (11)^2 + (14)^2 \right] (c) Midpoint Rule = 3[(3.5)2+(6.5)2+(9.5)2+(12.5)2]3 \left[ (3.5)^2 + (6.5)^2 + (9.5)^2 + (12.5)^2 \right]

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