Applications of Integration: Work and Volumes of Revolution

Applications of Integration

Work Required to Pump Water from Tanks

In calculus, integration can be used to compute the work required to move fluids, such as water, from tanks of various shapes. The work is calculated by integrating the force needed to move each infinitesimal layer of fluid over the distance it must be lifted.

• Work Formula for Fluids: The total work required to pump fluid from a tank is given by: $W = \rho g \int_a^b (h - y)A(y)dy$ where:

  • \(\rho\): Density of the fluid (kg/m3)

  • g: Acceleration due to gravity (m/s2)

  • h: Height to which the fluid is pumped (m)

  • y: Variable of integration representing the vertical position (m)

  • A(y): Area of the horizontal cross-section at height y (m2)

Example 1: Inverted Conical Tank

• Tank Shape: Inverted cone with height 9 m and base radius 2 m.

• Cross-sectional Area: At height y, the radius is \(r = \frac{2}{9}y\), so the area is: $A(y) = \pi \left(\frac{2}{9}y\right)^2 = \frac{4}{81}\pi y^2$

• Integral Setup: $W = \rho g \int_0^9 (9 - y) \frac{4}{81}\pi y^2 dy$

• Note: The constants \(\rho\) and \(g\) are not evaluated as per instructions.

Example 2: Spherical Tank

• Tank Shape: Sphere with radius 7 m, centered at the origin.

• Cross-sectional Area: At height y, the radius is \(x = \sqrt{49 - y^2}\), so the area is: $A(y) = \pi (49 - y^2)$

• Integral Setup: $W = \rho g \int_{-7}^7 (7 - y) \pi (49 - y^2) dy$

Volumes of Solids of Revolution

Integration is also used to find the volume of solids generated by revolving a region around a line. Two common methods are the shell method and the washer method.

• Shell Method: Sums up cylindrical shells formed by revolving vertical or horizontal slices.

• Washer Method: Sums up washers (disks with holes) formed by revolving slices perpendicular to the axis.

Example: Triangular Region R

Let R be the region bounded by \(y = 9 - x/2\), \(y = -1\), and \(y = 2x - 1\). The following are setups for the volume of the solid obtained by revolving R about various lines:

(a) About x = -2 (Shell Method)

• Volume: $\int_0^4 2\pi(x + 2)[(2x - 1) - (-1)]dx + \int_4^{20} 2\pi(x + 2)[(9 - x/2) - (-1)]dx$

(b) About x = -2 (Washer Method)

• Volume: $\int_{-1}^7 \pi[(18 - 2y) - (-2)]^2 - \pi\left(\frac{y + 1}{2} - (-2)\right)^2 dy$

(c) About y = -3 (Shell Method)

• Volume: $\int_{-1}^7 2\pi(y + 3)[(18 - 2y) - (\frac{y + 1}{2})]dy$

(d) About y = -3 (Washer Method)

• Volume: $\int_0^4 \pi[(2x - 1) - (-3)]^2 - \pi[(-1) - (-3)]^2 dx + \int_4^{20} \pi[(9 - x/2) - (-3)]^2 - \pi[(-1) - (-3)]^2 dx$

Additional info: The above setups demonstrate the use of both the shell and washer methods for finding volumes of revolution, a key application of integration in calculus.