Integration and solid of revolution

[kate45]

hi I'm super stuck with this question:

I'm super stuck with these two problems on one of my practice exams, can anyone help me out?

Find the integral between 4 and 3 of (u^2 + 1) / (u - 2)^2

and

Find the volume of the solid of revolution obtained by rotating, a full turn about the x-axis, the area between the x-axis and the curve y = 2 sin x, for x ∈ [ π , 3π ].

Thanks
if anyone can help me out

 

[rock.freak667]

Well for the first one


$$\int _3 ^{4} \frac{u^2 + 1}{(u - 2)^2} du$$


consider the integrand alone


$$\frac{u^2 + 1}{(u - 2)^2} \equiv \frac{u^2+1}{u^2-4u+4}$$


the degree of the polynomial in the numerator $\geq$ degree of the polynomial of the denommerator.

You need to divide it out until the degree of the polynomial in the numerator < degree of the polynomial of the denominator.

 

[Defennder]

For the second one just use the formula for evaluating the solid of revolution $$\pi \int^{3\pi}_{\pi} f(x)^2 dx$$