Review - Definite Integrals (end of Chapter 5)
Review - Definite Integrals (end of Chapter 5)
Review - Definite Integrals (end of Chapter 5)
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Calculus <strong>Review</strong> Worksheet<br />
(1) Evaluate each <strong>of</strong> the following integrals please.<br />
2<br />
4 2<br />
(a) ( )<br />
∫<br />
x + 4x dx<br />
(e)<br />
0<br />
2<br />
e<br />
∫<br />
e<br />
dx<br />
x ln x<br />
( )<br />
(b)<br />
∫<br />
2<br />
3x dx<br />
0 3 2<br />
7x<br />
−1<br />
(f)<br />
∫<br />
3<br />
4<br />
1<br />
2<br />
dx<br />
x 1<br />
−<br />
x<br />
π<br />
∫<br />
2<br />
(c) sin ( x) cos( x)dx<br />
− π<br />
4<br />
3<br />
(g)<br />
π<br />
∫<br />
2<br />
−<br />
π<br />
2<br />
( )<br />
cos x dx<br />
2<br />
( )<br />
1 + sin x<br />
π<br />
8<br />
∫<br />
2<br />
(d) tan ( 2x)dx<br />
− π<br />
8<br />
(h)<br />
∫<br />
2<br />
0<br />
2 2<br />
x 4 − x dx<br />
(2) Find the area under the graph <strong>of</strong><br />
x = 0 and x = 2.<br />
y = x<br />
2<br />
3x + 4 and above the x-axis, between<br />
(3) Find the area under one arch <strong>of</strong> the graph <strong>of</strong> y = cos( x)<br />
3 π and above the x-axis.<br />
3 2<br />
(4) Find the total area bound by the x-axis and the graph <strong>of</strong> y = x + 5x + 6x .<br />
(5) Find the area bound by the graph <strong>of</strong> 2 x + y = 2 and the coordinate axes.
Calculus <strong>Review</strong> Worksheet<br />
(6) Find '( x)<br />
F if<br />
F (x)<br />
=<br />
1<br />
∫sec<br />
( 2x)<br />
t<br />
2<br />
dt<br />
−1<br />
(7) Find the total area bound by each <strong>of</strong> the following curves please.<br />
(a)<br />
2<br />
y = x and y = x + 2<br />
(b)<br />
y<br />
2<br />
2 −<br />
= 6x − x and y = x 2x<br />
(c)<br />
2<br />
x = y and<br />
x<br />
=<br />
3<br />
−<br />
2<br />
2y<br />
(d) y = sin(x) , and y = sin(2x) if<br />
0<br />
≤<br />
x<br />
≤<br />
π<br />
2<br />
(e) y tan ( x)<br />
= , y = 0 , and<br />
x<br />
=<br />
π<br />
4<br />
(f)<br />
y<br />
3<br />
2<br />
= x + x − 2x , and y = − x + x + 6x<br />
3<br />
2<br />
−x<br />
(g) Calculator Problem y = 2 , and y = 3x 2 (Correct to 5 decimal places)<br />
2 −
Calculus <strong>Review</strong> Worksheet<br />
Answers<br />
256<br />
(1) (a) 15<br />
(e) ln(2)<br />
18<br />
(b) 7<br />
(f) 6<br />
π<br />
(c) 16<br />
3<br />
(g) 2<br />
π<br />
(d)<br />
1− π<br />
(h) π<br />
4<br />
(2) 9<br />
56<br />
(3) π<br />
6<br />
37<br />
(4) 12<br />
(5) 3<br />
2
Calculus <strong>Review</strong> Worksheet<br />
Answers<br />
(6) '( x)<br />
F = –2 csc(2x)<br />
(7) (a)<br />
9<br />
2<br />
(b)<br />
64<br />
3<br />
(c) 4<br />
(d) 2<br />
5<br />
(e)<br />
1<br />
−<br />
π<br />
4<br />
(f) 16<br />
(g) 4.28533