The base of a solid is the region in the xy-plane bounded by the curves y = e² 2.2x, y = = 2.2x² +0.8 and x = 1. Every cross-section of the solid perpendicular to the x-axis (and to the xy-plane) is a square. The volume of this object is: Feel free to use technology to evaluate the integral.

Write the given equation in Cylindrical coordinates. (x-95)^{2}+y^{2}=9,025 r=95 \sin (\theta) r=95 \cos (\theta) r=190 \sin (\theta) r=190 \cos (\theta)

\text { Evaluate the double integral } \iint_{\Omega}\left(x^{3}+2 y\right) d x d y \text { over the region } \Omega \text { given as in the diagram } below. (You need to work out the limits of the double integral first.)

Determine whether or not each of the following series converges. Show all of your work. \sum ^{\infty}_{n\mathop=1}=\frac{n^3-3n^2+1}{n^4+2n+1}

(a) Find a (to the nearest degree) in A ABC below. (b) Given that find tan(105°). Present your answer in fully simplified exact form. i) 2 partitions; ii) 5 function values. Give your answers to 3 decimal places. \tan (x+y)=\frac{\tan (x)+\tan (y)}{1-\tan (x) \tan (y)} \text { (c) Approximate } \int_{-4}^{4}\left(2^{x}+2^{-x}\right) d x \text { using the trapezoidal rule with }

Determine whether or not each of the following series converges. Show all of your work. \sum ^{\infty}_{n\mathop=2}\frac{-7}{n\ln\mleft(n\mright?^2}

(5) (5 marks) Let H be a solid hemisphere of radius a with constant density. (a) Find the centroid of H. (b) Find the moment of inertia of H about a diameter of its base.

Using full sentences, write a brief summary of the geometric transformation that occurs to the integral

During the past week of class Kota and Tyler explain they heard some terms they are to understand. In particular, they are struggling with what makes an integral improper and what it means for an improper integral to converge or diverge. In their textbook, they found the following problems in the section on improper integrals.

The base of a solid is the region in the xy-plane between the the lines y = x, y = 6x, x = 1 and x = 3. Cross-sections of the solid perpendicular to the x-axis (and to the xy-plane) are squares. The volume of this solid is: