Please show hand written work. I have difficulty reading typed ones. Also please try to explain the steps as much as possible for a clear understanding to your justification

Do all of the problems. Your grade will depend not only on the correctness of the answer, but on how well you justify it. Do not use a calculator, except for Question 1. 1. Use Riemann sums to estimate the centre ofmass of T's-ianitoba. No, really. The centre of mass exists, of course. There is a point on which one could balance the entire province. Use the formulae in your Statics textbook to gure out the technique that is used to nd the centre of mass and use Riemann sums to approximate the point. Use intervals of 100 km in width. Your mark will depend on (i) explaining the formula For the centre of mass, (ii) showing on a map the dimensions that you take, and (iii) showing the results and locating them in the province. 2. Recall queStion 3 from Assignment 1: (a) Let R be the region bounded hy the Function y = ex, the y axis, and the line y = E. Find its area. (b) Find the volume of the solid created if the region R is rotated about as : 1. (c) Supposing the volume from part (b) is empty, how much work is required to ll it with liquid pumped up from the x axis? Develop the formula for part (c) using Riemann sums. 3. For this question, we will use Riemann sums to evaluate the time required to move along a difcult path. Imagine a mountain climber moving down a mountain with the shape 3; = f(x) for U S x S I, where f(0) = it represents the peak of the mountain and fa} = U is at ground level (i.e., x) represents the outline of the x z crosssection of the mountain). The climber is moving at the speed 5306) along the mountainside (to, the climber's speed winning his} path is given by how far the}r are from the peak downswing. To nd how much time it takes For the climber to go from the peak to the bottom, nd reasonable assumptions For analyzing the path, use these to create Riemann sums that will approximate the time, and then create an explicit formula for the value