Math Snap

STEP 1

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

STEP 2

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

STEP 4

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

STEP 5

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

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