Calculus module 6 dbq

show work

1. Let be the three points of intersection, i.e. the solutions to . They are approximately

Then the area is

since over the interval we have , and over the interval we have .

2. Using the washer method, we generate washers with inner radius and outer radius . Each washer has volume , so that the volume is given by the integral

3. Each semicircular cross section has diameter . The area of a semicircle with diameter is , so the volume is

4. is continuous and differentiable everywhere, so the the mean value theorem applies. We have

and by the MVT there is at least one such that

for integers , but only one solution falls in the interval when , giving

5. Take the derivative of the velocity function:

We have when . For , we see that , while for , we see that . So the particle is speeding up on the interval and slowing down on the interval .

21

step-by-step explanation: