Download Advanced Mathematical Concepts

Advanced Mathematical Concepts
Chapter 15
Lesson 15-4
Example 1

Evaluate
3
3
x 2 dx .
1
The antiderivative of f(x) = x2 is F(x) = 3x3 + C.
3
3 x dx  3 x
1
2

3
3
 C 3
Fundamental Theorem of Calculus

3
 1  33  C   1   3  C 
 3
3

 9  9 or 18
Let x = -3 and 3 and subtract.
Example 2
Evaluate each indefinite integral.
a.
 6x dx
4
 6 x dx  6  5 x
1
4
5
C
 6 x5  C
5
b.
Constant Multiple of a Power Rule
Simplify.
  3 x  2 x  5 x  4 dx
1
 3x  2 x  5x  4 dx  3  4 x
3
3
2
2
 2  1 x3  5  1 x 2  4 x  C
3
2
3
2
5
4
3
2
 x  x  x  4x  C
4
3
2
4
Remember x1 = x and 4 = 4x0 .
Simplify.
Advanced Mathematical Concepts
Chapter 15
Example 3
Find the area of the shaded region.
The area is given by

1
1
( x3  2)dx .
1
The antiderivative of f(x) = -x3 + 2 is F(x) = -4x4 + 2x + C.
1
1 4
3
1   x  2 dx   4 x  2 x 1
1
+C is not needed with a definite integral.
4
4
  1 1  2 1   1  1  2  1
  4
 4

Let x = 1 and - 1 and subtract.
4
The area of the shaded region is 4 square units.
Example 4
As part of a sales promotion, a car dealership wants to put up an archway of balloons over a platform
that will showcase one of their cars. The arch is to be in the shape of a parabola. It will be 20 feet high
and 18 feet across the bottom. Using the point on the ground directly below the apex of the arch as the
origin, the equation of the arch is approximately
x2
y = 20 . How much area do they have under the arch in order to setup the platform and the car?
4.05
The area is given by

9
9
x dx .
20  4.05

2
9  20  4.05  dx  9  20  4.05 x
9

x2 
9
1
2
 dx
Rewrite the function.
9
 20 x  1  1 x3
4.05 3 9
Antiderivative; +C not needed
3
3
 20  9   1  9    20  9   1  9   Let x = 9 and - 9 and subtract.
12.15
12.15

 

 180  60    180  60  or 240
The area under the arch is 240 square feet.