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Math  /  Calculus

QuestionUse the following table to estimate ∫025f(x)dx\int_{0}^{25} f(x) d x. Assume that f(x)f(x) is a decreasing function. \begin{tabular}{c|c|c|c|c|c|c} \hlinexx & 0 & 5 & 10 & 15 & 20 & 25 \\ \hlinef(x)f(x) & 49 & 47 & 44 & 37 & 27 & 7 \\ \hline \end{tabular}
To estimate the value of the integral we use the left-hand sum approximation with Δx=\Delta x= □\square
Then the left-hand sum approximation is □\square
To estimate the value of the integral we can also use the right-hand sum approximation with Δx=\Delta x= □\square Then the right-hand sum approximation is □\square
The □\square of the left-and right sum approximations is a better estimate which is □\square

Studdy Solution

STEP 1

1. The function f(x) f(x) is decreasing.
2. We are using numerical integration methods (left-hand and right-hand sum approximations) to estimate the integral.
3. The interval [0,25][0, 25] is divided into equal subintervals based on the given table.

STEP 2

1. Determine Δx\Delta x.
2. Calculate the left-hand sum approximation.
3. Calculate the right-hand sum approximation.
4. Calculate the average of the left-hand and right-hand sum approximations for a better estimate.

STEP 3

Determine Δx\Delta x, the width of each subinterval. Since the x x values are 0,5,10,15,20,25 0, 5, 10, 15, 20, 25 , the subinterval width is:
Δx=5\Delta x = 5

STEP 4

Calculate the left-hand sum approximation. Use the function values at the left endpoints of each subinterval:
Left-hand sum=Δx×(f(0)+f(5)+f(10)+f(15)+f(20))\text{Left-hand sum} = \Delta x \times (f(0) + f(5) + f(10) + f(15) + f(20))
=5×(49+47+44+37+27)=5×204=1020= 5 \times (49 + 47 + 44 + 37 + 27) = 5 \times 204 = 1020

STEP 5

Calculate the right-hand sum approximation. Use the function values at the right endpoints of each subinterval:
Right-hand sum=Δx×(f(5)+f(10)+f(15)+f(20)+f(25))\text{Right-hand sum} = \Delta x \times (f(5) + f(10) + f(15) + f(20) + f(25))
=5×(47+44+37+27+7)=5×162=810= 5 \times (47 + 44 + 37 + 27 + 7) = 5 \times 162 = 810

STEP 6

Calculate the average of the left-hand and right-hand sum approximations for a better estimate:
Average=Left-hand sum+Right-hand sum2=1020+8102=915\text{Average} = \frac{\text{Left-hand sum} + \text{Right-hand sum}}{2} = \frac{1020 + 810}{2} = 915
The better estimate for the integral is:
915\boxed{915}

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