Volume Inverse Sec 036F49

1. **State the problem:** We need to evaluate the volume $V$ given by the integral $$V=\int_{24}^{55} \pi \left[12 - \left(3.8 \sec^{-1}(1.02x - 19.0957) + 6\right)\right]^2 dx.$$ 2. **Understand the formula:** This integral represents the volume of a solid of revolution or a volume defined by a function squared times $\pi$, integrated over $x$ from 24 to 55. The integrand is $$\pi \left[12 - \left(3.8 \sec^{-1}(1.02x - 19.0957) + 6\right)\right]^2.$$ 3. **Simplify the integrand:** Let $$f(x) = 12 - \left(3.8 \sec^{-1}(1.02x - 19.0957) + 6\right) = 6 - 3.8 \sec^{-1}(1.02x - 19.0957).$$ So the integrand is $$\pi [f(x)]^2 = \pi \left(6 - 3.8 \sec^{-1}(1.02x - 19.0957)\right)^2.$$ 4. **Rewrite the integral:** $$V = \pi \int_{24}^{55} \left(6 - 3.8 \sec^{-1}(1.02x - 19.0957)\right)^2 dx.$$ 5. **Approach to solve:** This integral involves the inverse secant function inside a quadratic expression. It is generally not elementary to integrate by hand. One would typically use numerical integration methods or software to evaluate this integral. 6. **Summary:** The volume $V$ is given by $$V = \pi \int_{24}^{55} \left(6 - 3.8 \sec^{-1}(1.02x - 19.0957)\right)^2 dx.$$ To find a numerical value, use numerical integration techniques such as Simpson's rule or computational tools.