📐 Comprehensive Integral Application Problem

Part A: Area Calculation (Definite Integral)

• A.1. Determine the coordinates of the intersection points of the two curves.
• A.2. Sketch the region $\mathbf{R}$. Identify which function represents the upper bound and which represents the lower bound within this region.
• A.3. Set up and evaluate the definite integral to find the exact area of the region $\mathbf{R}$.

Part B: Volume of Revolution (Disk/Washer Method)

The region $\mathbf{R}$ is revolved around the $\mathbf{x}$-axis.

• B.1. Set up the integral using the Washer Method to find the volume of the resulting solid.
• B.2. Evaluate the integral to find the exact volume of the solid of revolution.

Part C: Volume by Cross-Sections (Known Cross-Sections)

A solid is formed with the region $\mathbf{R}$ as its base. Cross-sections perpendicular to the $\mathbf{y}$-axis are $\mathbf{semicircles}$.

• C.1. Express the length of the diameter of a semicircular cross-section as a function of $\mathbf{y}$. (Hint: You may need to express $x$ in terms of $y$ for both functions).
• C.2. Set up the definite integral required to find the volume of this new solid.
• C.3. (Bonus/Challenge) Evaluate the integral set up in C.2.

Part D: Work (Lifting a Fluid)

Imagine a storage tank has the exact shape of the solid generated in $\mathbf{Part~B}$. The tank is full of a liquid with a weight density of $\mathbf{\rho}$ $\text{lb/ft}^3$.

• D.1. Set up the definite integral to find the total work required to pump all of the liquid out of the top of the tank.
• D.2. If the fluid is water ($\rho \approx 62.4 \text{ lb/ft}^3$), and the limits of integration are from $x=0$ to $x=1$ (assuming the region $R$ is scaled to represent a tank in feet), express the total work in $\text{ft-lb}$. (Do not evaluate the final integral, just provide the fully set-up integral with numerical bounds and constant $\rho=62.4$).