Download Fundamental Theorem Calculus

Calculus BC
Name:_______________________
Area Between Two Curves 4-D
Date: _______________________
For Problems 1 and 2, set up integrals that could be used to find the areas of the shaded regions. Do not
integrate. Show the equation(s) used to find the limits of integration for Problem 2 without using a
calculator.
y
1.
f ( x)  x
2
3
f(x)
g(x)
f(x)
2.
g ( x)   x2  2
(-1,1)
A=

1
1
x
f ( x)  x  4 x
g ( x)   x
2
(1,1)
______
For Problems 3 and 4, show equations used to find the limits of integration, show integral set ups for the
areas of the shaded regions, and then find the areas.
y
3.
No calculator.
f ( x)  3
y
4.
f(x)
g ( x)  4 x  x 2
Use a calculator.
f ( x)  2 x  x 2
x
g ( x)   3 x  1
x
For Problems 5 and 6, sketch regions bounded by the graphs of the given equations, show equations used to
find the limits of integration, show integral set ups, and find the areas.
5.
Use a calculator.
y  3t
y  t2
6.
No calculator. Hint: Write as
functions of y . (Isolate x )
1
2
y x
x  y2  3
7. Without using absolute value, why can’t the area of the region(s) bounded by y  4 x and
y   x 3 be written as a single integral using 2 and 2 as limits of integration? (Sketch graphs of
y  4 x and y   x3 in one coordinate plane before you answer this question.) Then use the
symmetry of the graphs to write a single integral for the area.
8. Without using a calculator, sketch a graph showing one of the regions bounded between
y  cos x and y  2  cos x . Then find the area of the region.
y
f(x)
9.
Use a calculator to find the shaded area between the curves
f ( x )  2 sin x and g ( x )  3cos x on [0,  ] as shown
at right. Show an integral set up and express your final
answer to 3 or more decimal place accuracy.
x

For Problems 10 through 17, sketch regions bounded by the graphs of the given equations, show equations
used to find the limits of integration, show integral set ups, and then find the areas.
10. The curve y  x 2  2 and the line y  2
11. The curve y  x 2 and the curve y  4 x  x 2
12. The curve y  x 3 and the curve y  3x 2  4
13. The curve y  x 2  4 x  5 and the curve y  2 x  5
14. The curve y  x 3 and the x-axis, from x = -1 to x = 2
15. The curve x  y 2 and the line x  y  2
16. The curve x  y 2 and the curve x  3  2 y 2
17. The curve x  y 3  y 2 and the line x  2 y
Use a calculator for Problems 18 & 19
18.
For f ( x)  ln x, find f   2 
19.
Find the area between f ( x)  e x and the x-axis on the interval 0, 2.
y
Use the graph of y  f ( x) at right to graph each
of the functions in Problems 15-17. Graph each
problem on a separate coordinate plane.
20.
y   f ( x)  1
21.
y  f ( x  1)
x

22.


y  2 f ( x)
Evaluate the expressions in Problems 23-26.
23.
d
dx

x
0
t dt
d
dx
24.

0
t dt
2x
25.
d
dx

x2
0
t dt
26.

x
0
t dt
For Problems 27 & 28, set up definite integrals that could be used to find the areas of the regions shown or
described. You do not need to evaluate the integrals that you set up.
27.
y
2
f ( x)  2 x 3

on  3,3
y = f(x)

x


28.
Region bounded by
g ( y)  y 2  1,
x  0, y  2, and
y 3
Without a calculator, sketch graphs and use geometry to evaluate Problems 29 and 30.
29.

4
0
3 x  2 dx
  2  x  dx
1
30.
2
31. A Spherical balloon is expanding at the rate of 5 cm3 /sec. How fast is the diameter of the
V 
balloon increasing when its volume is 36 cm3 ?
32.
If

2
0

a.

f ( x) dx  4 and f is an odd function, find:
0
2
f ( x) dx
d.
33.
4
 r3
3

b.

2
0
2
2
f ( x) dx
3 f ( x) dx
e.
c.
  f ( x)  1 dx
2

2
0
 f ( x ) dx
(Be careful on this one.)
0
Two men in a search party begin walking from Search Headquarters at the same time. One man walks
North at a rate of 4 ft/sec, while the other man walks West at a rate of 3 ft/sec. After both men have
walked for one minute, find
a. the distance separating the two men.
b. the rate at which the distance between the two men is changing.
Evaluate in Problems 17-19.
34.
37.
d
dx

2x
0
sin t dt
35.
d 5
dx x2

Find the value of this definite integral

sin t dt
36.
d 2t x
2 dx
dt t


4
0
tan x dx .
Evaluate the integrals in Problems 38-41 without using a calculator.
2
5
3 1
38.  x 3 dx
39.  2 dt
40.   x 2  2 x  ( x  1) dx
41.
2
1 t
42.
Use geometry to find

4
0
f ( x) dx
for the function shown at right.

 4  x,
f ( x)  
2

 4x  x ,
  dy

0 x2

2 x4

1
y
y 1
part of
a circle
y = f(x)







43. Use the f  and f  number lines below to sketch a possible graph of a continuous function f .



f  

0
2
44. . Use the graph of f  at right to sketch
a. a graph of f  .
b. a graph of f which passes through
the origin of the coordinate plane.
Use a separate coordinate plane for each
graph.



f  

-2
0

y




f  x
x



500 yd. fence