Suppose that we are asked to evaluate the integral . After performing the substitution , which of the following expressions evaluates to I?

Suppose we are asked to evaluate the following definite integral by substitution:  . Answer the following questions: What is your initial substitution u=? What is the lower limit of integration after your substitution? What is the upper limit of integration after your substitution? What is an exact value for I=?

Suppose that we are asked to find the area (A) between the curves     and    on the interval . Which of the following plots shows the shaded area between the two curves on ?

Suppose we are asked to evaluate the integral . If we set   and , which of the following is an expression for I in terms of u and v (NOT x)? Hints: For all real x:    For all real x:    Is the sine function even or odd?

Suppose that and   are fixed real numbers. Consider the following two limits (L and L2):    AND . Statement: The two limits are equal? (True/False)

Suppose we are asked to evaluate the following integral:  . Select the appropriate choices to complete the next equivalent expressions for I:  -   .

Extra Credit – Bonus! Suppose the f and g are both continuous and integrable functions on . Furthermore, say that we know that for all , .  We say that the “fractional part” of x, denoted by , is the real or non-integer part of x belonging to . Also, we have that for all x. For example, , and so on. Now suppose that we want an expression for the following integral that does not involve the function f: . If we set , enter two exact (integer) expressions for the numerator and denominator of I: Numerator of I=  Denominator of I=

Evaluate the following definite integral:   

Suppose that we want to evaluate the following integral:    Answer the following questions: Which u-sub to we choose first? u= What are the corresponding lower and upper limits of integration after the u-sub? and Which of the following methods do we use to evaluate the resulting integral after the substitution (type a capital single letter)? Integration by substitution Fundamental theorem of calculus Trigonometric substitution Partial fractions Integration by parts

4.2 Name each factor of production and their compensation. (4)