Difference between revisions of "009B Sample Midterm 1, Problem 1"

From Grad Wiki
Jump to navigation Jump to search
 
(24 intermediate revisions by 2 users not shown)
Line 1: Line 1:
−
<span class="exam">Evaluate the indefinite and definite integrals.
+
<span class="exam"> Let &nbsp;<math style="vertical-align: -5px">f(x)=1-x^2</math>.
  
−
::<span class="exam">a) <math>\int x^2\sqrt{1+x^3}dx</math>
+
<span class="exam">(a) Compute the left-hand Riemann sum approximation of &nbsp;<math style="vertical-align: -14px">\int_0^3 f(x)~dx</math>&nbsp; with &nbsp;<math style="vertical-align: 0px">n=3</math>&nbsp; boxes.
−
::<span class="exam">b) <math>\int _{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos(x)}{\sin^2(x)}dx</math>
 
  
 +
<span class="exam">(b) Compute the right-hand Riemann sum approximation of &nbsp;<math style="vertical-align: -14px">\int_0^3 f(x)~dx</math>&nbsp; with &nbsp;<math style="vertical-align: 0px">n=3</math>&nbsp; boxes.
  
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+
<span class="exam">(c) Express &nbsp;<math style="vertical-align: -14px">\int_0^3 f(x)~dx</math>&nbsp; as a limit of right-hand Riemann sums (as in the definition of the definite integral). Do not evaluate the limit.
−
!Foundations: &nbsp;  
 
−
|-
 
−
|1)
 
−
|-
 
−
|2)
 
−
|-
 
−
|-
 
−
|Answers:
 
−
|-
 
−
|1)
 
−
|-
 
−
|2)  
 
−
|}
 
  
−
'''Solution:'''
+
<hr>
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+
[[009B Sample Midterm 1, Problem 1 Solution|'''<u>Solution</u>''']]
−
!Step 1: &nbsp;
+
 
−
|-
+
 
−
|
+
[[009B Sample Midterm 1, Problem 1 Detailed Solution|'''<u>Detailed Solution</u>''']]
−
|-
 
−
|
 
−
|-
 
−
|
 
−
|-
 
−
|
 
−
|}
 
  
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Step 2: &nbsp;
 
−
|-
 
−
|
 
−
|-
 
−
|
 
−
|-
 
−
|
 
−
|-
 
−
|
 
−
|}
 
  
−
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
−
!Final Answer: &nbsp;
 
−
|-
 
−
|
 
−
|-
 
−
|
 
−
|}
 
 
[[009B_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Midterm_1|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 15:42, 12 November 2017

Let  .

(a) Compute the left-hand Riemann sum approximation of    with    boxes.

(b) Compute the right-hand Riemann sum approximation of    with    boxes.

(c) Express    as a limit of right-hand Riemann sums (as in the definition of the definite integral). Do not evaluate the limit.

Solution


Detailed Solution


Return to Sample Exam

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=009B_Sample_Midterm_1,_Problem_1&oldid=4045"