Integral Expression

[jlbc]

Let f be the function given by f(x)=3cosx. As shown above, the graph of f crosses the y-axis at point P and the x-axis at point Q.

The graph shown with the problem is the first quadrant of the graph of f(x)=3cosx. Point P exists at (0,3), and Point Q exists at (3.14/2, 0).

a) Write an equation for the line passing through the points P and Q.

I have already solved this portion; it is y= (-6/3.14)x+3.

b) Write an equation for the line tangent to the graph of f at point Q.

I have already solved this part; the answer is y=(-3)x+(3(3.14))/2

C) Find the x-coordinate of the point on the graph of f, between points P and Q, at which the line tangent to the graph of f is parallel to line PQ.

This value is 0.690.

This final part is the one I have trouble determining.

d) Let R be the region in the first quadrant bounded by the graph f and the line segment PQ. Write an integral expression for the volume of the solid generated by revolving the region R about the x-axis. Do not evaluate.

I do not understand integrals at all. Thank you for your help.

 

[galactus]

d) Let R be the region in the first quadrant bounded by the graph f and the line segment PQ. Write an integral expression for the volume of the solid generated by revolving the region R about the x-axis. Do not evaluate.

I do not understand integrals at all. Thank you for your help.

Here is a graph of the region you are revolving. I can not give a lesson on integration here, but you use what is known as a solid of revolution using the washer method.

It comes from the area of a circle. When you figure the area of an infinite number of circles, or infinitely thin washers, and put them all together, it gives the volume of the region.

You know how you find the area of a washer by subtracting the inner area(where the hole is) from the outer area?

\(\displaystyle {\pi}(R_{2}^{2}-R_{1}^{2})\)

Well, same concept here.

\(\displaystyle {\pi}\int_{0}^{\frac{\pi}{2}}\left[(3cos(x))^{2}-(\frac{-6}{\pi}x+3)^{2}\right]dx\)

This is the region between f(x) and the line PQ revolved around the x-axis. Thus, giving volume.

This is one of the coolest and most useful topics in calc. Get to know it and you can find the volume of just about anything.

 

[jlbc]

Thank you!