Question

Find the volume of the solid generated by rotating the region of the x-y plane between the line y=7, the curve y=4sin(x)+3 for -pi /2 11 years ago

Answer 1

You can use the disk method to solve for the volume. I've attached a picture of the region bounded by the two functions (it's shaded in light green).

The function that represents the radius of the disk of interest would be represented by the equation \(7-\left(4\sin(x)+3\right)\).
How would we get this equation? You can think of subtracting two whole regions from each other to get the region in between. Hopefully this makes sense:
Shade the entire region below \(y=7\). From this, subtract the entire region below \(4\sin(x)+3\), and what you are left with is the region between \(y=7\) and \(4\sin(x)+3\), the region of interest to us.

Volume is just \(\text{Area} \times \text{Length}\). We know that the area of the disk is given by \(\text{Area} = \pi r^2\); the \(\text{Length}\) term is just the \(dx\) term of the integral that we'll integrate:

\[\int\limits_{-\pi/2}^{5\pi/2} \pi \left(7-\left(4 \sin(x)+3\right) \right)^2 \ dx\]
Pulling out the \(\pi\), which is a constant term, and simplifying, we should get
\[\pi\int\limits_{-\pi/2}^{5\pi/2} \left( 16 \sin^2(x) - 32 \sin(x) + 16\right) \ dx\]
Take the integral. You should know how to take the integral of trig functions already. \(\sin^2(x)\) can be integrated by either converting it to an equivalent identity or using a table of integrals.

Answer 2

thanks