Math

Question Find the Left and Right Riemann sums, then use them to find the Trapezoidal sum on the interval [3,7][-3,7] given the table of xx and f(x)f(x) values.

Studdy Solution

STEP 1

Assumptions
1. The table provides discrete values of a function f(x)f(x) at specific points xx.
2. We are interested in finding the Left Riemann sum, Right Riemann sum, and the Trapezoidal sum over the interval [3,7][-3,7].
3. The interval is partitioned at the points x=3,0,2,6,7x = -3, 0, 2, 6, 7.
4. The Left Riemann sum uses the function value at the left endpoint of each subinterval.
5. The Right Riemann sum uses the function value at the right endpoint of each subinterval.
6. The Trapezoidal sum is the average of the Left and Right Riemann sums.

STEP 2

First, we will calculate the Left Riemann sum. This involves multiplying the function value at the left endpoint of each subinterval by the width of the subinterval and summing these products.

STEP 3

Identify the subintervals and their widths. The subintervals are [3,0][-3,0], [0,2][0,2], [2,6][2,6], and [6,7][6,7]. The corresponding widths are 33, 22, 44, and 11.

STEP 4

Calculate the Left Riemann sum using the function values at the left endpoints of the subintervals.
L=f(3)3+f(0)2+f(2)4+f(6)1L = f(-3) \cdot 3 + f(0) \cdot 2 + f(2) \cdot 4 + f(6) \cdot 1

STEP 5

Substitute the function values from the table into the Left Riemann sum formula.
L=13+32+54+61L = 1 \cdot 3 + 3 \cdot 2 + 5 \cdot 4 + 6 \cdot 1

STEP 6

Compute the Left Riemann sum.
L=3+6+20+6=35L = 3 + 6 + 20 + 6 = 35

STEP 7

Next, we will calculate the Right Riemann sum. This involves multiplying the function value at the right endpoint of each subinterval by the width of the subinterval and summing these products.

STEP 8

Calculate the Right Riemann sum using the function values at the right endpoints of the subintervals.
R=f(0)3+f(2)2+f(6)4+f(7)1R = f(0) \cdot 3 + f(2) \cdot 2 + f(6) \cdot 4 + f(7) \cdot 1

STEP 9

Substitute the function values from the table into the Right Riemann sum formula.
R=33+52+64+(3)1R = 3 \cdot 3 + 5 \cdot 2 + 6 \cdot 4 + (-3) \cdot 1

STEP 10

Compute the Right Riemann sum.
R=9+10+243=40R = 9 + 10 + 24 - 3 = 40

STEP 11

Finally, we will calculate the Trapezoidal sum. The Trapezoidal sum is the average of the Left and Right Riemann sums.
T=L+R2T = \frac{L + R}{2}

STEP 12

Substitute the values of the Left and Right Riemann sums into the formula for the Trapezoidal sum.
T=35+402T = \frac{35 + 40}{2}

STEP 13

Compute the Trapezoidal sum.
T=752=37.5T = \frac{75}{2} = 37.5
The Left Riemann sum is 3535, the Right Riemann sum is 4040, and the Trapezoidal sum is 37.537.5.

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