The plate R in Figure 8, bounded by the parabola y=x^2 and...

The plate $R$ in Figure $8,$ bounded by the parabola $y=x^{2}$ and $y=1,$ is submerged vertically in water (distance in meters). $$ \begin{array}{c}{\text { (a) Show that the width of } R \text { at height } y \text { is } f(y)=2 \sqrt{y} \text { and the fluid }} \\ {\text { force on a side of a horizontal strip of thickness } \Delta y \text { at height } y \text { is }} \\ {\text { approximately }(\rho g) 2 y^{1 / 2}(1-y) \Delta y .} \\ {\text { (b) Write a Riemann sum that approximates the fluid force } F \text { on a side }} \\ {\text { of } R \text { and use it to explain why }} \\ {F=\rho g \int_{0}^{1} 2 y^{1 / 2}(1-y) d y} \\ {\text { (c) Calculate } F .}\end{array} $$

The plate $R$ in Figure $8,$ bounded by the parabola $y=x^{2}$ and $y=1,$ is submerged vertically in water (distance in meters).
$$
\begin{array}{c}{\text { (a) Show that the width of } R \text { at height } y \text { is } f(y)=2 \sqrt{y} \text { and the fluid }} \\ {\text { force on a side of a horizontal strip of thickness } \Delta y \text { at height } y \text { is }} \\ {\text { approximately }(\rho g) 2 y^{1 / 2}(1-y) \Delta y .} \\ {\text { (b) Write a Riemann sum that approximates the fluid force } F \text { on a side }} \\ {\text { of } R \text { and use it to explain why }} \\ {F=\rho g \int_{0}^{1} 2 y^{1 / 2}(1-y) d y} \\ {\text { (c) Calculate } F .}\end{array}
$$

Key Concepts

Fluid Pressure on Submerged Surfaces

Fluid pressure increases with depth and is given by the product of the fluid density, gravitational acceleration, and the depth below the surface. This pressure acts perpendicularly on every point of a submerged surface, resulting in a net force that depends on the area and orientation of the surface.

Differential Elements and Slicing Method

This method involves dividing a continuous object into small, manageable pieces—often thin slices or strips—whose individual contributions to a quantity, like force, can be easily calculated. In problems involving fluid forces, each slice is analyzed independently, and its force is determined using properties like the width of the object at a given depth and the pressure acting on it.

Riemann Sums and the Concept of Approximation

Riemann sums provide a way to approximate the total effect of a variable quantity by summing the contributions from many small segments. By partitioning an interval into subintervals, calculating an approximate value for each, and adding them, one can estimate the overall integral, which in the limit as the segments shrink, becomes exact.

Definite Integration for Aggregating Continuous Effects

Definite integration is the process of summing the contributions of infinitesimally small elements to find a total effect or value over an interval. It provides a precise calculation of cumulative quantities, such as the total fluid force on a surface, by integrating a function that represents the differential contribution across the entire domain.

Geometric Modeling of Physical Boundaries

In many physical applications, such as calculating forces on submerged surfaces, it is essential to model the geometry of the object accurately. By expressing the shape of the boundary with mathematical functions, one can derive expressions—such as the width at a given height—that are used in setting up integrals to calculate forces or other physical quantities.

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