Note that you need to study Lesson 4(The Method of u Substitution), Lesson 5 (Area between Two Curves) and Lesson 6 (Volumes) from my Calculus 2 book to prepare for this assignment. I think you should find this assignment fairly straightforward if you do thoroughly study and do all the Practise Problems I give you in these lessons.
Don't have my book? You can download a sample containing two lessons here (unfortunately, the sample contains lessons 1 and 11):
Question 1 is integration obviously. Hint: you will use u substitution for some but not all of these three integrals. If you do use u sub to solve part (c), make sure you don't actually use u because that variable is already in the problem. You will have to use a different letter.
Question 2 is more of the same, but this time they are definite integrals. Some people change the endpoints to u values if they are using u substitution to solve a definite integral. I do not advise this. I suggest, that you write down the indefinite integral first and solve it. Then, return to the definite integral and sub the endpoints into your solution.
Some key facts you should know to properly compute the definite integrals in this problem:
elna = a will come into play. Watch out for the chance to use log laws too.
ln ab = b lna
Note also that if you are computing the definite integral from -a to a of an odd function, the integral is zero. Click this link for more info:
Also note that cosine is an odd function since cos(-x) = -cos(x).
You will need to prove that the function given in question 2(d) is an odd function.
Question 3 is not an integral problem. You cannot use methods of integration to solve it. First, separate the problem into two different integrals by splitting the 3 term away from the square root term. Now draw a graph of each of these two functions for the region between -2 and 2. (Draw a graph of y = 3 and a separate graph of y = square root of (4 - x2).) You should now be able to compute the area between the graph and the x-axis from x = -2 to x = 2 in each case using basic geometry (area of a rectangle, area of a circle).
Question 4 is like my questions in Lesson 5. Make sure you do draw the required graph and carefully identify the region to shade and compute. You will need to separate the problem into more than one integral, like I do in my question 2. Take each pair of the three curves and set them equal to each other to find the points of intersection. Keep in mind that you are told that x > 0, so do not shade any regions where x is negative.
