Volume of water in a swimming pool A rectangular swimming pool is 30 ft wide and 50 ft long. The

step 1 In this problem, we have a swimming pool, which is defined, and the height of the swimming pool

Volume of water in a swimming pool A rectangular swimming...

Volume of water in a swimming pool A rectangular swimming pool is 30 $\mathrm{ft}$ wide and 50 $\mathrm{ft}$ long. The accompanying table shows the depth $h(x)$ of the water at $5-\mathrm{ft}$ intervals from one end of the pool to the other. Estimate the volume of water in the pool using the Trapezoidal Rule with $n=10$ applied to the integral $$V=\int_{0}^{50} 30 \cdot h(x) d x$$

Volume of water in a swimming pool A rectangular swimming
pool is 30 $\mathrm{ft}$ wide and 50 $\mathrm{ft}$ long. The accompanying table shows
the depth $h(x)$ of the water at $5-\mathrm{ft}$ intervals from one end of the
pool to the other. Estimate the volume of water in the pool using
the Trapezoidal Rule with $n=10$ applied to the integral
$$V=\int_{0}^{50} 30 \cdot h(x) d x$$

Key Concepts

Volume Estimation Using Integration

Volume estimation through integration involves computing the volume of a solid by integrating the cross-sectional area along a particular dimension. This approach applies the fundamental concept of accumulation, where the integral sums up infinitely many thin slices or segments of the object to yield the total volume.

Trapezoidal Rule

The Trapezoidal Rule is a numerical integration method that approximates the area under a curve by dividing the region into trapezoids rather than rectangles. By calculating the area of these trapezoids and summing them, this technique provides an estimate of the definite integral, and it is particularly useful when function values are known at discrete points.

Definite Integral

A definite integral represents the accumulated sum of a function's values over a given interval, which is often interpreted as the total area under the curve between two limits. This concept is fundamental in calculus for calculating quantities that accumulate continuously, such as area, mass, work, and volume.

Numerical Integration

Numerical integration involves approximating the value of a definite integral when an exact analytical solution is difficult or impossible to obtain. This is typically achieved by evaluating the function at specific points and using techniques that sum contributions over small subintervals to estimate the total area.

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