question 1 For our first example, consider the functions j(x)=x26x+9 and k(x)=x1, and suppose we're asked to "find the area enclosed by j(x) and k(x)." If we look at the graphs of these two functions:

in finding the "area enclosed by j(x) and k(x)", we want need to find the area of the green shaded region. We would like to use Theorem 44, but first we need to find the bounds of integration, i.e. aa and bb. In these types of problem, they are not explicitly given, rather we need to determine them by determining where the two functions intersect. To determine where two functions intersect, e.g. what points (x,y) appear on both functions, we set them equal to each other, i.e. j(x)=k(x) x26x+9=x1 Moving everything on the right side of the equal sign to the left side, we get: _______=0 which has two solutions (found via factoring): x=_______ (smallest) and x=_______ (largest)

Question 2 Outside of finding an antiderivative by finding a function whose derivative is the original function, we will only learn one other method or tool for evaluating antiderivatives: Substitution (sometimes called u-substition). The first half of Calculus II focuses on several other tools for evaluating antiderivatives. The method of integration by substitution is the antiderivative analog to the chain rule in differentiation, which can be seen in the following theorem statement from APEX Caclulus UND Edition: For example, if we consider the function h(x)=sin(4x3+3x2) then h(x)=cos(4x3+3x2)(12x2+6x)

Part 2 of 11

Thus, we know that cos(4x3+3x2)(12x2+6x)dx= _____________________________ + C