Prepared: Sat Dec 4 13:29:00 CST 1999 Reprepared: Wed Dec 15 12:29:58 CST 1999

MATH 1271

QUIZ IX

1) Evaluate the following indefinite integrals.

displaymath189

Use substitution tex2html_wrap_inline191 , then tex2html_wrap_inline193 and

eqnarray14

displaymath195

Use substitution tex2html_wrap_inline197 , then tex2html_wrap_inline199 and

eqnarray36

displaymath201

Use substitution tex2html_wrap_inline203 , then tex2html_wrap_inline205 and

eqnarray54

2) Evaluate the following definite integrals.

displaymath207

Use substitution tex2html_wrap_inline209 , then tex2html_wrap_inline211 and

eqnarray68

Since we have a definite integral we have to evaluate this expression at the upper limit and then subtract from that the value of the expression at the lower limit:

displaymath213

displaymath215

Use substitution u=x/2 -1, then du=dx/2 and

eqnarray93

Since we have a definite integral we have to evaluate this expression at the upper limit and then subtract from that the value of the expression at the lower limit:

displaymath221

displaymath223

Use substitution tex2html_wrap_inline225 , then tex2html_wrap_inline227 and

eqnarray123

Since we have a definite integral we have to evaluate this expression at the upper limit and then subtract from that the value of the expression at the lower limit:

displaymath229

3) Sketch the graph of tex2html_wrap_inline231 . (You DON'T need to show each step of finding the critical points, increasing/decreasing intervals, etc.) Find the area of the region bounded by the graph of f and the x-axis (i.e. the region below the x-axis and above the graph.)

displaymath239

Either by using the graph or factoring the function we find out that the zeros of f(x) are -1 and 5. So the area will be (attention to the negative sign in front of f)

eqnarray149

4) Apply the Fundamental Theorem of Calculus to find the derivative of

displaymath245

i.e. find f'(x). Please justify your answer.

The Fundamental Theorem of Calculus states:

Suppose f is a continuous function on [a,b]. If F is defined on [a,b] by

displaymath249

then F'(x)=f(x) for all x in [a,b].

Here the function inside the integral, namely tex2html_wrap_inline255 is continuous everywhere, in particular on the interval [1,b], for any b. So the theorem applies giving the result

displaymath257

for all x.

5) Consider the function tex2html_wrap_inline261 on the interval [1,5]. Divide the interval into 4 subintervals, all of the same length. Write the coordinates of tex2html_wrap_inline263 . (How many tex2html_wrap_inline263 's will you have?) Write the Riemann sum (in symbols) for the left-end/right-end/mid-points of the intervals. Then write the Riemann sum as a sum of the numeric values.

(Solution will not be added later unfortunately. The TA is extremely busy nowadays.)


Quiz 8